




















,V </> 






INFALLIBLE LOGIC 



A VISIBLE AND AUTOMATIC 
SYSTEM OF REASONING 



- / 



THOMAS D. HAWLEY 

OF THE CHICAGO BAR 




*i^V£^ 



ROBERT SMITH PRINTING COMPANY 

LANSING. MICHIGAN 

1896 



2 



.v\^ 



Of CONG*"*' 
WASMINS2E 



Entered according to Act of Congress, in the year 1896, 

By THOMAS D. HAWLEY, 

In the Office of the Librarian of Congress, at Washington. 



TABLE OF CONTENTS. 



t 

INTRODUCTION. 
SECTION. PAGE. 

1. This work is for Lawyers, etc 1 

2. My Object 1 

3. What Part is Original 1 

4. A New System of Reasoning 1 

5. Thinking is Made Visible 1 

6. Beginners and Advanced Logicians 1 

7. A Graphic and Diagrammatic System 2 

8. Object of the System 2 

9. Contradictory Propositions 2 

10. Inductive Reasoning 2 

11. Difficult Problems 2 

12. Advice to the Reader 2 

CHAPTER I. 

LOGIC. 

13. Definition of Logic 3 

14. Sir William Hamilton on Reasoning 3 

15. New Facts Not Discovered by Logic 4 

16. Prima Facie Meaning of a Proposition 4 

17. Language 4 

18. Ideas 4 

19. Law of Opposites 5 

20. Positive and Negative 5 

21. Making Combinations 5 

22. Elimination 6 

23. The Old Logic 6 

24. Arithmetical Reasoning 6 

25. Mathematical Reasoning 6 

26. Utility of Logic 7 

27. Grammar 7 

CHAPTER II. 

DEFINITIONS. 

28. Logic, a Technical Science S 

29. Nominal and Real Definitions 8 

30. Accidental Definitions 8 

31. Limits^of Definitions 8 

32. Logical Definitions 9 



i v TABLE OF CONTENTS. 

SECTION. PAGE. 

33. Physical Division 9 

34. Classification 9 

35. Names 9 

36. Adjectives 10 

37. Definite Sentences 10 

38. Existence 10 

39. Singular Names 10 

40. General Names 10 

41. Concepts H 

42. Individual Names H 

43. Predicables H 

44. Relatives 12 

45. Negative Names 12 

46. Categorematic Words 12 

47. Collective Names 12 

48. Judgment is Naming 12 

49. Propositions 13 

50. Terms 13 

51. Complex Terms 13 

52. Positive and Negative Terms 13 

53. Contraries 13 

54. Universals and Particulars 13 

55. Affirmatives and Negatives 14 

56. Quantity and Quality 14 

57. Particular Affirmatives 15 

58. Universal Negatives 15 

59. Particular Negatives 15 

60. Aflirmo and Nego 15 

61. Indesignate Subjects 16 

62. Distributed and Undistributed Terms 16 

63. Meaning of Undistributed 16 

64. Distributed, and Undistributed, Illogical 16 

65. Predication 17 

66. Universal Affirmatives 17 

67. Logical Form 17 

68. Conversion of Universal Affirmatives 18 

69. Particular Affirmatives 18 

70. Universal Negatives 18 

71. Particular Negatives 18 

72. Copulative Propositions 18 

74. Exclusive Propositions 19 

75. "Unless" and "Except" 19 

76. Elliptical Language 19 

77. Real Propositions 19 

78. Verbal Propositions 19 

79. Formal Propositions 19 



TABLE OF CONTENTS. V 

SECTION. PAGE. 

80. True and False Proposit ions 20 

81. Generalization 20 

82. General Terms 20 

83. Abstract Names 21 

84. Concrete Names 21 

85. "Extension" 21 

86. Connotative Names 21 

87. Kant's Divisions 21 

88. Hypothetical Propositions 22 

89. Disjunctive Propositions 22 

CHAPTER III. 

THE LAWS OF THOUGHT. 

90. An Infallible Logic 24 

91. Law of Relativity 24 

92. Law of Opposites 24 

93. Law of Identity 25 

94. Formula for the Law of Identity 25 

95. Certainty 26 

96. The Law of Contradiction 26 

v^97. Law of the Excluded Middle 27 

98. Formula of Law of Excluded Middle 27 

99. Law of the Excluded Middle, a Corollary 27 

100. Law of Logical Division 27 

101. Dichotomy 29 

/102. The Law of Elimination 31 

103. The Reasoning Process 31 

104. Development of a Proposition 31 

105. Positing Opposite Terms 32 

106. Law of Combinations 33 

107. Development of Combinations 33 

CHAPTER IV. 

INFERENCE. 

108. Immediate Inference 35 

109. Inference, the Result of a Process 35 

110. Inference by Conversion 35 

111. Synonymous Propositions 36 

1 12. Inference by Qualification 36 

CHAPTER V. 

SIGNS. 

113. Use of Signs 37 

114. Sign for the Copula :>,- 

115. Sign for "or" 37 



vi TABLE OF CONTENTS. 

SECTION. PAGE. 

116. Signs for Terms 37 

117. No Sign for Conjunctions 37 

118. Compound Subjects 38 

119. Copula, Not Necessary in Logic 38 

120. Signs for Names 39 

121. Distinctions of the Old Logic 39 

122. No Sign for a Denial . 39 

123. Order of Reading Letters t 39 

124. Stating Alternatives 39 

125. Universe of Discourse 40 

126. Sections, Rows and Files 41 

127. Diagram Representing the Universe of Discourse 42 

128. A Reasoning Frame 42 

129. The Brain is a Thinking Machine 42 

130. An Example, A is B 42 

131. Consistent Propositions : 43 

132. An Example, Salt is Chloride of Sodium 43 

133. An Example, Chloride of Sodium is Salt 45 

134. Two Negative Propositions 46 

135. Negative Terms 47 

136. An Example What is Not Chloride of Sodium is Not Salt. . . 48 

137. Testing Simple Propositions 48 

138. Key to our System 49 

139. Mechanical Reasoning 49 

140. Letters in the Sections 49 

141. Law of Combinations 49 

142. Diagram for One Term 49 

143. Law of Identity 50 

144. Law of Contradiction represented 50 

145. Law of the Excluded Middle represented 50 

146. Diagram for Two Terms 50 

147. Law of Relativity 50 

148. Law of Opposites 50 

149. Law of Identity 50 

150. Law of Contradiction 51 

151. Law of the Excluded Middle 51 

152. Reasoning Frame, a Foundation for Logic 51 

153. Example, Death is not Life 51 

154. Example, Life is Not Death 52 

155. A Short Method of Working Examples 53 

156. Example, Not— Life is Death 54 

157. Reasoning, a Uniform Process 55 

158. Interpreting Negative Terms 55 

159. Prof. Venn on Diagrams 55 

160. Boole's System 56 

161. Eulerian System 56 



TABLE OF CONTENTS. vii 

SECTION. PAGE 

162. Kant's Diagrams 56 

163. Bolzano's Parallelograms 56 

164. Dr. Urqnand's Diagrams 56 

165. Discovery of this Method 56 

166. Ploucquet's Squares 56 

167. Prof. Venn's Ellipses 57 

168. Inconsistency in the Premises 57 

169. Use of Eliminated Combinations 58 

170. Inconsistent Premises 58 

171. Aristotle on Singular Terms 58 

172. Exercises for Practice 58 

CHAPTER VI. 

INDEFINITE PROPOSITIONS. 

173. Indefinite Propositions 59 

174. Predicates of Common Terms 59 

175. Subject and Predicate are Names 60 

176. Universal Affirmative Propositions 60 

177. Boole's Sign for "Some" 60 

178. Letters in Two Combinations 61 

179. Conclusions of Special Value 61 

180. Inferring Some from All 61 

181. Universal Negative Propositions 62 

182. Reading the Subject 62 

183. Particular Affirmative Propositions 62 

184. Function of Logic 62 

185. Exclusion of Few, Many, Most 63 

186. Particular Negative Propositions 63 

CHAPTER VII. 

SIMPLE CATEGORICAL PROPOSITIONS INVOLVING THREE TERMS. 

187. Combinations from Three Terms 64 

188. Ways of Making Diagrams 64 

189. Diagram for Three Terms 65 

390. Diagram for Three Terms 66 

191. Diagram for Three Terms 66 

192. Changing the Letters 66 

193. Lettering the Sections (57 

194. Reading the Combinations 67 

195. Reading the Combinations 67 

196. Omitting Letters 68 

197. Reading the Letters 68 

198. Repeating the Letters (58 

199. Reading the Conclusions 68 

200. Omitting the Middle Term 68 



viii TABLE OF CONTENTS. 

SECTION. PAGE. 

201. Example, A Man is a Rational Animal 69 

202. Patent Meanings of Propositions TO 

203. Eliminatng Inconsistent Combinations 71 

204. Deduction 71 

205. Induction 71 

206. Deduction and Induction, Reverse Operations 73 

207. Negative Propositions 73 

208. Example, What is Not Fit to Learn, etc 73 

209. Example, London is the Capitol of England 75 

210. Example, The Substance of Least Density is Hydrogen 77 

211. Example, What is not the Substance of Least Density 78 

212. Reading Propositions Backward 78 

213. Alternative Propositions are Indefinite 79 

214. An Example of an Inconsistent Proposition 80 

215. Inconsistency in the Premises 82 

216. Law of Opposites 82 

217. Test of Inconsistency 82 

218. Example of an Inconsistent Proposition 82 

219. Examples for Practice 84 

CHAPTER VIII. 

INDUCTION. 

220. A Proposition is two or more Names 85 

221. The Problem of Inductive Logic 85 

222. What Premises will Produce Given Conclusions 85 

223. How to Define Letter-Terms 85 

224. Rule for Obtaining Definitions 88 

225. Rule for Obtaining the Premises 88 

226. Reading Propositions Backward 89 

227. Prof. Jevons on Induction 90 

228. Definitions of Positive and Negative Letters 91 

229. Prof. Jevons on Differentiation 91 

230. Self-Contradiction in the Premises 91 

231. An Example in Induction 91 

232. The Tentative Method 93 

233. An Example in Induction 94 

234. An Example in Induction 95 

235. Rule for Finding the Premises 90 

236. An Example in Induction 96 

237. Example, "The Tenth Amendment" 97 

238. Producing all the Premises 99 

239. Difference Between Induction and Deduction 99 

240. Premises and Conclusions 100 

241. Reasoning is Obtaining Other Propositions 100 

242. Reasoning Can Not Make Any Progress 100 

243. Equivalent Propositions 10) 



TABLE OF CONTENTS. ix 

SECTION. PAGE. 

244. An Idea and its Opposite 100 

245. The Problem of Logic 100 

246. The Problem of Logic 100 

247. The Universe of Discourse 101 

248. The Tenth Amendment 101 

249. The Powers Reserved to the States 103 

250. The Powers Reserved to the People 104 

251. The Powers Delegated to the U. S 105 

252. Prof. Jevons on Induction 100 

253. Dr. Keynes' Solution of the Inductive Problem 107 

254. Prof. Jevons on Combinations of Eight Things 107 

255. Prof. Jevons on Combinations of Four Terms 107 

256. Inductive Problems Containing Four or More Terms 107 

257. Miss Jones on Inductive Fallacies 108 

258. Induction makes clear only 108 

259. Two Problems in Reasoning 108 

260. Equivalent Propositions 110 

261. Prof. Jevons' Table 110 

262. An Example from Prof. Jevons Ii2 

263. Two Examples from Prof. Jevons 113 

264. An Example from Prof. Jevons 115 

265. Exercises for Practice 118 

CHAPTER IX. 

A CHAIN OF REASONING. 

266. Inferences from True Propositions 119 

267. Inferences from False Propositions 119 

268. Reasoning from the Truth of a Proposition 119 

269. "The Tenth Amendment" 119 

270. Stating Propositions 119 

271. Eighteen Chains of Reasoning 125 

272. An Exhaustive View 125 

273. Possible Readings 125 

274. Reasoning from the Truth of a Proposition 120 

CHAPTER X. 

TERMS. 

275. Single Terms 127 

276. Combination of Single Terms 127 

277. Expression of Disjunctive Terms 127 

278. Determinants 127 

279. Alternants 127 

280. Expression of Alternants 127 

281. Reading the Determinants 128 

282. Meaning of Disjunctives 128 



X TABLE OF CONTENTS. 

SECTION. PAGE. 

283. Meaning of Disjunctives 128 

284. What is the Contradictory of AB? 128 

285. Terms have no Contradictories 128 

286. Reading Eliminated Combinations 128 

287. Reading Uneliminated Combinations 128 

288. Finding the Contradictory of AB 128 

289. Finding the Opposite of a or b 129 

290. The Opposite of A or B 129 

291. The Opposite of ab 129 

292. The Opposite of A or BC 130 

293. The Opposite of ab or ac 131 

294. The Opposite of ABC or ABI) 131 

295. The Opposite of a or b or cd 132 

296. Complex Terms 133 

297. Logical Contraries 133 

298. An Example 134 

299. Prof. Keynes on Obversion 134 

300. An Equivalent f or AB or AC 134 

301. An Equivalent for A or BC 135 

302. An Equivalent for A or aB 136 

303. An Equivalent for AC, or AD, or BD 136 

304. Meaning of Bb 137 

305. Meaning of B or b 137 

306. Exercises for Practice 137 

CHAPTER XI. 
ELIMINATION. 

307. Elimination, a Process 138 

308. Meaning of Elimination 138 

309. Results of Elimination 138 

310. A Supposition Lies Behind a Premise 138 

311. Premises Cause Elimination 139 

CHAPTER XII. 

EXAMPLES CONTAINING THEEE TERMS. 

312. Granite is not a Sedimentary Rock 140 

313. All Planets are Subject to Gravity 141 

314. "He that is of God Heareth My Words" 142 

315. John is a Man 143 

316. All Avaricious Men Refuse to Give Money 144 

317. A Science which Furnishes the Mind, etc 146 

318. Mount Blanc is the Highest Mountain in Europe 147 

319. Sodium is a Metal 149 

320. Neptune is a Planet 150 

321. Whales are not True Fish 151 



TABLE OF CONTENTS. xi 

SECTION. PAGE. 

322. All Heated Solids Give Continuous Spectra 152 

323. All Fixed Stars are Self -Luminous 153 

324. Some Metals are of less Density than Water 154 

325. Exercises for Practice 156 

CHAPTER XIII. 

DISJUNCTIVES. 

326. Disjunctive Propositions 158 

327. Disjunctives Imply Opposition 158 

328. Law of Identity Applies to Disjunctives 158 

329. Prof. Jevons on Disjunctives 158 

330. Prof. Jevons on "Or" 159 

331. Prof. Venn on Contradictories 160 

332. Logical Meaning of Elimination 160 

333. Solids or Liquids or Gases, etc 160 

334. A or B = C or D 163 

335. ab = cd 165 

336. cd = ab 165 

337. Appearance of Equivalent Propositions 166 

338. Propositions Resemble Boxes 166 

339. Obtaining Equivalent Propositions 167 

340. Making Thoughts Visible 167 

341. A or B = C or D 167 

342. A New Method 167 

343. Reduction of Alternatives to Categoricals 169 

344. Diagrams, at first blank Diagrams 170 

345. Obtaining Alternative Definitions 171 

346. Reduction of Alternatives 171 

347. Wealth is what is transferable, etc 171 

348. Rule for Combining Alternatives 174 

349. Gems are either rare Stones, etc 174 

350. Red Colored Metal is either Copper or Gold 176 

351. Abscissio Inflniti 178 

352. Example, The Members of a Board, etc 179 

353. Implication of Disjunctives 180 

354. Disjunctives are mostly Fallacies 180 

355. Ambiguities 181 

356. Mistakes to Avoid 181 

357. Stating Alternatives 181 

358. Alternants are Exclusive 182 

359. Either B or C Exists 182 

360. Either the Witness is Perjured, etc 183 

361. Modus Pouendo Tollens 184 

362. Exclusi veness of "Or" 185 

363. Modus ToUendo Ponem 185 

364. A = B or C = D 186 



xii TABLE OF CONTENTS. 

SECTION. PAGE. 

365. Two Disjunctive Premises 186 

366. Pure Alternative Syllogisms 188 

367. What Propositions Deny 189 

368. Every Blood-vessel is, etc 190 

370. Two Negative Propositions 193 

371. Example, a or b is c or d 195 

372. Example, a or b is c or D 196 

373. Example, a or B = C or D 197 

374. Reduction of Disjunctives to Categoricals 198 

375. Example, A or B or C = D 198 

376. Example, a or b or c = D 200 

377. Example, a or b or c = D or E 201 

378. Example, A = B or C = D 202 

CHAPTER XIV. 

OR. 

379. Or 204 

380. Prof. Hamilton on Disjunctives 204 

381. Prof. Venn on the meaning of "Or" 204 

382. Meaning of "Or" 204 

383. Meaning of "Or" 204 

384. St. Aquinas on "Or" 205 

385. Prof. Boole on "Or" 205 

386. Prof. Jevons on "Or" 205 

387. Meaning of "Or" 205 

388. Meaning of "Exclusive" 205 

389. Miss Jones on "Or" 205 

390. Stating "Or" 206 

391. Or is Indefinite 206 

392. Exercises for Practice 206 

CHAPTER XV. 

HYPOTHETICAL PROPOSITIONS. 

393. Hypotheticals Indicate Doubt 208 

394. Hypotheticals are Indefinite 208 

395. Conversion of Hypotheticals 208 

396. Suppressed Premises in Hypotheticals 208 

397. Example, The Case of Caesar is the Case of an Usurper 208 

398. Example, Caesar was an Usurper 209 

399. Example, If Caesar was an Usurper 210 

400. Hypotheticals and Conditionals 211 

401. Example, If A = b, then A = c 211 

402. Example, If A = B, then A = C 212 

403. Example, If A = B then A = C 313 

404. Example, If A = b then A = c 214 



TABLE OF CONTENTS. xiii 

SECTION. PAGE. 

405. Example, If A = B, then A = C 214 

406. Example, If the First Preachers of the Gospel, etc 215 

407. Example, If the Education of Certain Children, etc 216 

408. Example, If the Weather Continues Fine, etc 217 

409. Example, If He Caught the Infection He will Die 218 

410. Example, If Force is Expended, etc 219 

411. Example, If this River has Tides, etc 220 

412. Example, If this River has Tides, etc 221 

413. Value of Hypothetical 222 

414. Prof. Venn's Illustration 222 

415. Miss Jones on Conditionals 223 

416. Hypotheticals are Universals 223 

417. Disjunctives which are Reciprocating 223 

418. Example, If A is B, then A is C 223 

419. Substitutes for "If" 224 

420. Hypotheticals are Compound 224 

421. Quality of Hypotheticals. ., 224 

422. Contradictory Hypotheticals 224 

423. Conversion of Hypotheticals 225 

424. Example, If a straight line, etc 226 

425. Geometrical Problems 227 

426. Example, If A is true, then C is true 227 

427. Mr. McColl on Hypotheticals 228 

428. Mr. Welton on Hypotheticals 229 

429. Inconsistent Consequents 238 

CHAPTER XVI. 

HYPOTHETICAL PEOPOS1TIONS CONTINUED. 

430. Inconsistent Antecedents 239 

431. Example, A = B or C = D 240 

432. Example, A or B = C or D 242 

433. Example, If A, then C 244 

434. Example, If this pen is not cross-nibbed, etc 245 

435. New names for Propositions- 246 

436. Identifying Propositions 247 

437. Example, If Patience is a Virtue 247 

438. Example, If a Righteous God, etc 248 

439. Example, If A is true, etc 248 

440. Example, If A = B, then C = D 249 

441. Example. If Water is Salt, etc 250 

442. Example, Whenever C is I), etc 252 

443. Example in Darapti 253 

444. Example, Never When C is D, etc 254 

445. Example, If A = B, C = D 255 

446. Example, If A = b, then C = I) 255 

447. Example, If A = B, then C = d 256 



xiv TABLE OF CONTENTS. 

SECTION. PAGE. 

448. Example, If A = b, C = d 257 

449. Example, If A = B, C = D 258 

450. Example, If A = B. C = D 259 

451. Exercises for Practice 200 

CHAPTER XVII. 

DILEMMAS. 

452. Dr. Keynes on Dilemmas 202 

453. Prof. Bain on Dilemmas 202 

454. Example, If the Barometer Falls, etc 202 

455. Example, If the Barometer Falls, etc 203 

450. Example, If the Barometer Falls, etc 2(>4 

457. Example, If the Barometer Falls, etc 265 

458. Example, If the Barometer Falls, etc 200 

459. Example, If the Barometer Falls, etc 267 

400. Example, If the Orbit of a Comet, etc 268 

401. Example, If a Classical Education, etc 269 

462. Example, If Schoolmasters Can Claim, etc 270 

403. Example, If ^Eschines Joined, etc 272 

404. Example, If a Science Furnishes, etc 273 

405. Example, If This Man Were Wise, etc 274 

4G0. Example of a Simple Destructive I Mlenima 275 

407. Methods for Finding Combinations to be Eliminated 277 

408. Example, If A = B, C ■ = D and E = F 278 

409. Prof. Bain on Dilemmas 270 

470. Example, If E = f , A = b 270 

471. Example, If a King of Spain, etc 280 

472. Equivalent Disjunctives 282 

473. Exercises for Practice 282 

CHAPTER XVIII. 

STATING PROPOSITIONS. 

474. Examples of Stating Propositions 284 

CHAPTER XIX. 

READING. 

475. Examples in Reading 286 

476. Diagram for One Term 286 

477. Diagram for Two Terms 287 

478. Diagram for Two Terms 287 

479. Diagram for Three Terms 288 

480. Diagram for Four Terms 288 

481. Diagram for Four Terms 289 

482. Diagram for Four Terms 289 



TABLE OF CONTENTS. XV 

SECTION. PAGE. 

483. Diagram for Four Terms 290 

484. Diagram for Four Terms 290 

485. Diagram for Four Terms 291 

486. Diagram for Four Terms 291 

487. Diagram for Four Terms 292 

488. Diagram for Four Terms 293 

489. Diagram for Four Terms 293 

490. Diagram for Four Terms 294 

491. Diagram for Four Terms 294 

492. Diagram for Four Terms 295 

493. Diagram for Four Terms 295 

494. Diagram for Six Terms 296 

495. Diagram for Six Terms 297 

496. Diagram for Six Terms 298 

497. Diagram for Six Terms 299 

498. Diagram for Four Terms 299 

499. Diagram for Four Terms 300 

500. Example, a or D = B or C 300 

CHAPTER XX. 

THE SYLLOGISM. 

501. Syllogism Contains Three Terms 302 

502. Syllogism Contains Three Propositions 302 

503. Syllogisms are Divided into Four Figures 302 

504. The First Figure .302 

505. The Second Figure 302 

506. The Third Figure 302 

507. The Fourth Figure 303 

508. Syllogisms are Divided into Four Forms 303 

509. Universal Affirmatives 303 

510. Universal Negatives 303 

511. Particular Affirmatives 303 

512. Particular Negatives 303 

513. Affirmo and Nego 303 

514. Conclusions from the Figures 303 

515. Order of Subject and Predicate 303 

516. The Second Figure 304 

517. The Third Figure 304 

518. Six Rules of the Syllogism 304 

519. The Premises 304 

520. Example. All Horned Animals Ruminate 304 

521. Dictum dv oinni et nullo 304 

522. Rules of the Syllogism 305 

523. Example. All Horned Animals Ruminate 305 

524. Reducing the Predicate 306 



xv i TABLE OF CONTENTS. 

SECTION. PAGE. 

525. The Reasoning Process 308 

526. Example, No Savages Have the Use of Metals 308 

527. Example, Some Europeans are Englishmen 310 

528. Example, Cornishmen are Englishmen 311 

529. Prof. Bain on Syllogisms 312 

530. Syllogisms are Useless 313 

531. Prof. Bain on the Syllogism 313 

532. Syllogism, not Fallacious 313 

533. Miss Jones on the Syllogism 314 

534. Four Terms to the Syllogism 314 

535. Lotze on the Syllogism 314 

536. Independent Premises 314 

537. Reading Eliminated Combinations 316 

538. Prof. Venn on the Syllogism 317 

539. Useless Distinctions 318 

540. The Booleian System 318 

541. Lotze on the Syllogism 318 

542. Pretended Syllogisms - 320 

543. Prof. Bain on Illicit Process 321 

544. Example of Illicit Process 322 

545. Example of Illicit Process 323 

546. Prof. Bain on Negative Premises 323 

547. Lotze on the Syllogism 325 

548. Miss Jones on the Syllogism 325 

549. Miss Jones, the Word "Some" 326 

550. Miss Jones, the Syllogism 326 

551. Subject and Predicate are Names 327 

552. Miss Jones, the Terms in a Syllogism 327 

553. Subject and Predicate Must be Identical 327 

554. Subject and Predicate, Names for the Same Thing 327 

555. Miss Jones' Criticism on Jevons 328 

556. Miss Jones' Criticism on Jevons 329 

CHAPTER XXI. 

THE FIGUKES. 

557. Figures of the Syllogism 330 

558. First Figure 330 

559. Second Figure 331 

560. Third Figure 331 

561. Fourth Figure 332 

562. Tables of the Figures 333 

563. Dr. Keynes on the Figures 333 

564. Particular Propositions 334 

565. Sixteen Combinations of Premises 334 

566. Valid Combinations of Premises 334 



TABLE OF CONTENTS. xvii 

SECTION. PAGE. 

567. Dr. Keynes on Figure One , 334 

568. Abseissio Infiniti 335 

569. Example of the Third Figure 336 

CHAPTER XXII. 

THE MOODS. 

570. The Moods of the Syllogism 337 

571. The Eleven Allowable Moods 337 

572. Six Moods to a Figure 337 

573. Example, All Human Creatures are Entitled to Liberty 337 

574. Five Neglected Moods 338 

575. Nineteen Valid Moods 338 

576. Moods of the First Figure 339 

577. Celerant 339 

578. Darii 340 

579. Ferio 341 

580. Cesare 342 

581. Camestres 343 

582. Celerant 345 

583. Festino 345 

584. Baroco 346 

585. Darapti 347 

586. Disamis 348 

587. Datisi 349 

588. Felapton 350 

589. Bocardo 351 

590. Ferison 352 

591. Bramantip 353 

592. Camenes 354 

593. Dimaris 355 

594. Fesapo 356 

595. Fresison 358 

596. Rules for the Moods of the Second Figure 359 

597. Inferring "Some" from ''All" 359 

598. Subaltern Moods 359 

599. Strengthened Syllogisms 359 

600. Informal Syllogisms 360 

601. Mnemonic Lines 361 

602. Meaning of the Mnemonic Lines 361 

603. What the Letter "s" Means 361 

604. What the Letter "p" Means 361 

005. What the Letter "m" Means 362 

606. What the Letter "k" Means 3(52 

607. What the Capital Letters Mean 362 

608. The Galenian Figure 362 



xviii TABLE OF CONTENTS. 

SECTION. PAGE. 

009. Thompson's Criticism of the Galenian Figure 362 

010. Enthymernes 363 

611. Enthymernes of the Second Order 363 

<?12. Enthymernes of the Third Order 303 

013. Examples of Enthymernes 303 

014. Polysyllogisms 363 

015. Prosyllogisms 303 

010. Epicheiremas 303 

617. Sorites 365 

618. Aristotelian and Gloclenian Sorites 305 

019. Example of the Aristotelian Sorites 305 

(520. Example of the Glocenian Sorites 305 

021. Rules of the Sorites 305 

022. Sorites in Figures Two and Three 366 

623. Sorites in Baroco and Bocardo 300 

024. Special Rules of the Sorites 360 

625. Example of a Sorites 367 

020. Sorites Yielding Particular Conclusions 308 

627. Examples of Informal Syllogisms :'. ,- > s 

628. Whately's Claim for t he Syllogism 368 

629. Ray's Claim for the Syllogism 368 

630. Spaulding's (Maim for the Syllogism 308 

031. Mill's Claim for the Syllogism 360 

CHAPTER XXIII. 
PROPOSITIONS. 

632. A Proposition is an Act of the Judgment 370 

633. Propositions Have Two Terms :IT» » 

034. "Not" is a Part of the Predicate 370 

035. When Propositions are Opposed 370 

030. Contraries 370 

037. Subalterns 370 

038. Sub-contraries 370 

039. Subalterns Differ in Quantity , 370 

640. Sub-contraries Cannot Both be False 371 

041. Subalterns May Both be True or False 371 

042. If One Contradictory is True, the Other is False 371 

043. Conversion 371 

044. Conversion of Universal Negatives 371 

045. Conversion of Particular Affirmatives 371 

640. Conversion of Particular Negatives 371 

047. Conversion of Universal Affirmatives 371 

048. When Propositions are not True 372 

019. When Propositions are Distrit uted 372 

050. Propositions are True or False 372 



TABLE OF CONTENTS. xix 

SECTION. PAGE. 

651. Predicate is Another Name for the Subject 372 

652. The Copula in Relation to Time 373 

653. "Is," not Equational 373 

654. Order of the Premises 373 

655. Two Singular Premises 374 

656. Prof. Bain on Negation 374 

657. Prof. Jevons on Negative Propositions 374 

65$. No Conclusions from Indefinite Premises 374 

659. Example from Prof. Jevons 374 

660. Substitutes for "not" 375 

661. New Names for Propositions 375 

662. Terms A and a are Opposites 375 

663. Negative Equivalents 375 

664. Inferences 376 

665. What is an Inference 376 

066. Example of a Negative Equivalent 378 

CHAPTER XXIV. 

QUANTIFIC ATION. 

667. "All" and "Some" 379 

668. Sir Wm. Hamilton on Quantification 379 

669. Four New Forms 379 

670. Six Forms of Propositions 380 

671. Dr. Keynes on Quantification 380 

672. Meaning of the Word "Some" 380 

673. Prof. Lotze, on Quantification 380 

674. Subject and Predicate are Names 383 

675. Criticism on Boole's System 383 

676. A Particular Copula 384 

677. Miss Jones on Quantification 384 

678. Difference Between Subject and Predicate 385 

G79. Miss Jones on Quantification 385 

680. Example of Quantification 385 

681. Miss Jones on Converting an E Proposition 385 

682. Reading the Word "No" 386 

683. Miss Jones on the Theory of Inclusion : 387 

6S4. Sir Wm. Hamilton's Eight Forms 388 

685. Comprehensiveness of This System 389 

686. Sir Wm. Hamilton's Position 389 

6S7. Dr. Bain on Quantification 389 

688. Dr. Keynes on Predication 389 

689. Dr. Keynes on the Eight Forms 390 

690. Dr. Keynes, I. U. N 390 

691. Dr. Keynes on A. Y. 1 391 

692. Example of A, Y, 1 392 



XX 



TABLE OF CONTENTS. 



SECTION. PAGE. 

693. Dr. Keynes on the Proposition "n" 392 

694. Dr. Keynes Criticism of Thompson 393 

695. Force of the Proposition "n" 393 

696. Dr. Keynes, A, U, Y, n 394 

697. Example of Y and n 394 

698. Example of U and n 395 

699. Dr. Keynes on the Proposition "w" 396 

700. Exclusive Propositions # 396 

701. Stating the Eight Forms 396 

CHAPTER XXV. 

INCONSISTENCY. 

702. The Sign of Contradictoriness 397 

703. If A is AB? 397 

704. If No A is B? 397 

705. If AB is AB? 397 

706. If Ab is Ab? 397 

707. If A is AB is False? 398 

708. If No A is B is False? 398 

709. If AB is AB is False? 398 

710. If Ab is Ab is False? 398 

711. When a Denial is Equivalent to an Affirmative 398 

712. Ambiguity in Propositions 398 

713. When Proposition are Not True 398 

714. Conversion of Contradictories 399 

715. Equivalent for All A is All B 399 

716. Equivalent for A is AB 399 

717. Equivalent for a = ab 399 

718. Equipollent Propositions 399 

719. Inferring "Some from All" 399 

720. When Propositions are Inconsistent 399 

721. Method of Finding Inconsistent Propositions 400 

722. Example of Inconsistent Propositions 401 

723. Example, The Tenth Amendment 402 

724. A Method of Obtaining Inconsistent Propositions 406 

CHAPTER XXYI. 

CONVERSION. 

725. Identical Propositions ! 407 

726. Order of Reading Combinations 407 

727. Convertend and Converse 407 

728. Rules for Conversion 407 

729. Conversion of E 408 

730. Conversion of A 408 

731. Conversion of I .« 408 



TABLE OF CONTENTS. xx i 

SECTION. PAGE. 

732. Aristotle's Conversion of E 408 

733. Dr. Keynes' Conversion of A 409 

734. Dr. Bain on the Value of "Some" 409 

735. Example of Contraposition 409 

736. Example of Conversion 409 

737. Conversion by Negation 410 

738. Example, Every True Poet is a Man of Genius 410 

739. Meaning of "May," -Can," etc 411 

740. Example, A Virtuous Man Cannot Betray His Country 411 

741. Stating of Propositions 411 

742. Prof. Bain on Obversion 412 

743. Obversion of A 412 

744. Obversion of 1 412 

745. Obversion of E 412 

. 746. Obversion of 412 

747. Trof. Bain's Rule for Obversion 412 

748. Prof. Lotze, Inference from Negation 413 

749. Prof. Lotze on the Denial of A 413 

750. Prof. Lotze on the Denial of 414 

751. Prof. Lotze on the Denial of 1 414 

752. Prof. Lotze on Impure Conversion 415 

753. Prof. Lotze, Reciprocal Judgments 415 

754. Prof. Lotze on the Conversion of 1 416 

755. Prof. Lotze on the Conversion of 416 

756. Miss Jones, Terminology for Conversion 417 

757. Subversion 417 

758. Obversion 418 

759. Reversion 418 

760. Introversion 418 

761. Contraversion 418 

762. Contraversion of 419 

763. Retroversion 419 

764. Examples of Conversion 419 

765. Inversion 419 

766. Coincidental Propositions 420 

767. What the Old Logic Says 420 

768. Making Subjects and Predicates Equivalent 420 

769. Dr. Keynes on the Conversion of A, E, I, 421 

770. Example of Equivalent Propositions 422 

771. Example of Contradictory Propositions 423 

772. Dr. Keynes on Complementary Propositions 424 

773. Dr. Keynes on Obversion 424 

774. Example of Obversion 424 

775. Example of Obversion 425 

776. Example of Obversion 42.~> 

777. Example of Obversion 42*; 



xxii TABLE OF CONTENTS. 

SECTION. PAGE, 

778. Other Names for Obversion 426 

779. Formal and Material Obversion 426 

780. Material Obversion is Illogical 427 

781. Dr. Keynes on Contraposition 427 

782. Contraposition Has Two Forms 427 

783. Example of Contraposition 427 

784. Obverted Contrapositive 428 

7S5. Rule for Obtaining the Contrapositive 428 

786. Example of Contraposition 428 

787. Example of Contraposition 429 

788. Dr. Keynes' Quotation from De Morgan 42$ 

789. Dr. Keynes' Criticism of Jevons 429 

790. Dr. Keynes on Inversion 430 

791. Inverse and Obverted Inverse 430 

792. Example of Inversion 430 

793. Example of Inversion 431 

794. Dr. Keynes' Methods of Conversion 431 

795. Example of Inversion 431 

796. Inferring a Proposition with Not- A for a Subject 432 

797. Dr. Keynes on Inversion 433 

798. Examples of the Different Kinds of Conversion 434 

799. Additional Examples of Conversion 435 

800. Example of the Different Kinds of Conversion 435 

S01. Additional Examples 436 

802. Example, Some Dogs are All Fugs 436 

803. Example, Salt is Chloride of Sodium 437 

804. Examples from Whately's "Elements of Reasoning" 438 

805. Examples in Camestres 440 

806. Example in Baroco 441 

807. Example in Bocardo 443 

808. Reductio ad Impossible 444 

809. Miss Jones on the Conversion of Hypotheticals 446 

810. Miss Jones' Examples 447 

811. Example, If Honesty is not the Best Policy, etc 448 

CHAPTER XXVII. 

ELIMINATION. 

S12. Dr. Keynes on Obversion 449 

813. Example from "Formal Logic" 449 

814. Example from "Formal Logic" 450 

815. Example of Equivalent Propositions 451 

816. Example of Equivalent Propositions 451 

817. Example of Equivalent Propositions 452 

818. Example of Equivalent Propositions 453 

819. Example of Equivalent Propositions 454 



TABLE OF CONTENTS. xxiii 

SECTION. PAGE. 

820. True and False Propositions 455 

821. Alexander of Aphrodisias 456 

822. Indirect Contraposition 457 

CHAPTER XXVIII. 

LOGICAL EXISTENCE. 

823. Dr. Keynes' Suppositions 458 

824. Both Subject and Predicate Imply Existence 458 

825. Logic Concerned Only with Words and Thoughts 458 

826. Logic Has Nothing to do With Existence 459 

827. Total Elimination of a Letter-Term 459 

828. No Predication Possible for Non-existing Things 459 

829. The Universe of Discourse 459 

830. Logic Not Interested with Actual Existences 459 

831. Dr. Keynes' Position 460 

832. Miss Jones' Position 460 

CHAPTER XXIX. 

NUMERICAL REASONING. 

833. Logic Cannot Solve Numerical Problems 461 

834. Example of Numerical Reasoning from "Formal Logic". . . . 461 

835. Example from Dr. Keynes 461 

836. Example from "Formal Logic" 462 

837. Example from "Formal Logic" 463 

538. Examples from "Formal Logic" 464 

CHAPTER XXX. 

COMPLEX PROPOSITIONS. 

539. Examples of Complex Propositions 466 

840. Example from "Formal Logic" 466 

841. An Example of Contradictories 467 

842. Example, AB = AC or DE 468 

843. An Example of an Inference 469 

844. An Example of Non-Equivalents 470 

845. An Example of Equivalents 471 

846. An Example of Non-Equivalents 472 

S47. Example, Given A or B Equals CD 472 

848. An Example, Given CD = ABCD 473 

849. An Example of Non-Equivalents 474 

850. Example, A = BC or DE 475 

851. An Example of Inconsistents 476 

852. Examples from "Formal Logic" 476 

853. Example from "Formal Logic" 477 

854. Example from "Formal Logic" 477 



xxiv TABLE OF CONTENTS. 

SECTION. PAGE. 

855. Example from "Formal Logic" 478 

856. Method for Solving Problems 480 

857. An Example of Consistents 480 

858. An Example of Non-Equivalents 481 

859. An Example of Consistents 482 

860. An Example of Consistents 483 

861. An Example of Non-Equivalents 483 

862. Example, AB = ABC 484 

863. An Example of Inference 485 

864. An Example of Non-Equivalents 485 

865. Example of Equivalent Propositions 486 

866. An Example of Equivalents 487 

867. Example, F = AB or bee 488 

868. Example of Equivalents 489 

869. Example of Inference 490 

870. Kinds of Propositions 491 

871. Kinds of Consistent Propositions 491 

872. Kinds of Inconsistent Propositions 491 

873. Inference and Inf erend 491 

874. Method for Finding Whether Propositions are Consistent .... 491 

875. Method for Finding Whether Propositions are Consistent .... 491 

876. Method for Finding Whether Propositions are Equivalent. . . 491 

877. Inf erend and Inference 491 

878. When Propositions are Merely Consistent 492 

879. Method of Finding Inconsistent Propositions 492 

880. Method of Finding Inconsistent Propositions 492 

881. Propositions, when Contradictories 492 

882. Propositions, when Perfect Contradictories 492 

883. An Example of Inference 492 

884. Example, A = B or C 493 

885. Example of Equivalents 494 

886. Example of Inference 495 

887. Example, Wherever the Properties A and B are Combined. . 496 

888. A Problem in Elimination 497 

889. A Problem in Inference 498 

CHAPTER XXXI. 

EXAMPLES. 

890. The Murder Problem 500 

891. The Steam Yacht Problem 501 

892. The Young Ladies Problem 503 

893. The Fishing Trip Problem 504 

894. The Hunting Trip Problem 505 

895. The Common Council Problem 506 

896. The Tenth Amendment 508 



TABLE OF CONTENTS. xxv 

SECTION. PAGE. 

897. Problem by Prof. Jevons 510 

898. Example from Jevons 511 

899. Example from Jevons 512 

900. Problem from Prof. De Morgan 512 

901. Problem about a Class of Things 513 

902. The Warm-Blooded Vertebrates Problem 514 

903. Problem about Buildings 51G 

904. Problem, If a Nation has Natural Resources, etc 518 

905. Problem, When the Properties A and B are Missing 520 

906. The Same Problem 522 

907. Problem from "Symbolic Logic" 522 

908. Problem, Equivalent Propositions 523 

009. Problem from "Formal Logic" 526 

910. The Scientific Society Problem 528 

911. Example, "He that Believeth," etc 531 

912. Example, "Except a Man be Born of Water" 533 

CHAPTER XXXII. 

INDUCTIVE EXAMPLES. 

913. Example in Induction 535 

914. Example in Induction 535 

915. Example in Induction 538 

916. Example in Induction 539 

917. New Method of Finding Equivalents 543 

918. Finding Categorical Equivalents 544 

919. Example in Induction 546 

920. Example in Induction 548 

921. Example in Induction 550 

922. Example in Induction 551 

923. Example in Induction 553 

924. Example in Induction 554 

925. Another Method of Finding Equivalents 556 

926. Example in Finding Equivalents 557 

927. Example in Finding Equivalents 557 

928. Example in Finding Equivalents 559 

929. Example in Finding Equivalents 559 

930. Example in Finding Equivalents 560 

931. Example in Finding Equivalents ' 560 

932. Finding Contradictory Propositions 562 

933. Another Method of Finding Contradictories 565 

934. Example in Finding Contradictories 566 

935. Example in Finding Contradictories 567 

936. Different sets of Contradictories can be found 569 

937. Example in Finding Contradictories 569 

938. Example in Finding Contradictories 571 



xxvi TABLE OF CONTENTS. 

SECTION. PAOE. 

939. Example in Finding Contradictories 573 

940. Example in Finding Contradictories 574 

941. Example in Finding Contradictories 576 

942. Example in Finding Contradictories 577 

943. Example in Finding Contradictories 579 

( .»44. Finding Alternative Equivalents 580 

945. Example, "The Tenth Amendment" 581 

946. The Method of this System 584 

947. Eulerian Circles, Useless for Complex Propositions 584 

948. Negative Terms arc on a par with Affirmatives 584 

949. Conclusions, so many Modes of Expression 584 

950. The Old Logic, Limited to Three Terms 585 

951. This System Abolishes Uncertainty 585 

952. This System Tuts Off Debate 585 

953. A New Method of Finding Contradictories 585 

CHAPTEB XXXV. 

I'AI.I.M IKS. 

954. Fallacy a False Reasoning 586 

955. Examples of Fallacies 586 

956. Fallacy of Equivocation 587 

957. Fallacy of Reasoning in a Circle 588 

958. Fallacy of Begging t he Question 588 

959. Fallacy of Self-Contradiction 589 

960. Fallacy of irrelevant Conclusion 589 

961. Fallacy of Argumentwn ad Hominem 589 

962. Fallacy of Contusion of Ideas 589 

963. Fallacy of Suppression of Truth 590 

964. Example of a Fallacy 590 

965. Example of a Fallacy 591 

966. Example of a Fallacy 592 

967. Example of a Fallacy 593 

968. Example of a Fallacy 594 

969. Example of a Fallacy 594 

970. Example, He who thrusts a knife 595 

971. Dr. Bain on Fallacies 595 

972. Dr. Bain on Relativity 595 

973. Dr. Bain on Kant's Autonomy 596 

974. Prof. Venn, on the Doctrine of Fatalism 596 

975. Fallacies of Relativity 596 

976. Irrelevant Question 596 

977. Putting more Questions than One, as One 597 

978. Examples from Prof. De Morgan 597 

979. De Morgan's Most Difficult Problem 597 

980. De Morgan, on the Word "All" 598 



TABLE OF CONTENTS. xxvii 

SECTION. PAGE. 

981. Example, What You Bought Yesterday, etc 598 

982. The Imperfect Dilemma 600 

983. The Sophism of Diodorus Cronus 600 

CHAPTER XXXVI. 

UTILITY OF LOGIC. 

984. Extravagant Claims of the Old Logic 601 

-985. Logic Will Not Discover New Facts 601 

986. Common Sense 601 

987. Reason, the Crowning Glory of Man 601 

988. Reasoning, the Foundation of all Knowledge 601 

989. The Advantages of Our System 602 

990. The Utility of Our System 602 

CHAPTER XXXVII. 

PKOBLEMS. 

991-1053. Exercises for Practice 603 

APPENDIX. 

HISTORICAL NOTES. 

1054. Zeno of Elea 607 

1055. Aristotle 607 

1056. Epicurus 607 

1057. Antisthenes 607 

1058. The Stoic Doctrine 608 

1059. Aristotle 608 

1060. Proclus 608 

1061. The Socratic Logic 608 

1062. Sextus Empericus 608 

1063. The Schoolmen 609 

1064. The Disputes of the Schoolmen 609 

1065. The Nominalists 609 

1066. The Realists 609 

1067. The Conceptualists 609 

1068. Particular and Universal Ideas 610 

1069. Opponents of the Scholastics 610 

1070. Roscellinus 610 

1071. Pierre Abelard 611 

1072. Raymond Lully 611 

1073. William Occam 612 

1074. Peter. Ramus 612 

1075. Thomas Hobbes 612 

1076. Jacques Benigne Bossuet 613 

1077. John Locke 613 



xxviii TABLE OF CONTENTS. 

SECTION. PAGE. 

1078. Christian Wolff 614 

1079. Gottfried Ploucquet 614 

1080. Denys Diderot 614 

1081. Etienne Bonnet de Condillac 615 

1082. George Campbell 615 

1083. Lord Karnes 615 

1084. L. H. Wagner 615 

1085. George W. F. Hegel 616 

1086. Ventura 616 

1087. Sir Wm. Hamilton 616 

1088. Dr. Thomas Brown 617 

1089. John Stuart Mill 617 

1090. George Boole 618 

1091. Archbishop Bichard Whately 618 

1092-1095. Brewster's Encyclopedia 620 

1096-1101. Victor Cousin 621 

1102-1111. Spencer's Principles of Psychology 623 

1112-1121. Logic, Encyclopedia Britannica 625 

1122-1155. Augustus De Morgan 629 

1156-1157. Bain's Deductive and Inductive Logic 638 

1158-1159. Krause's Vocabulary of Philosophy 639 

1160. Logic, Brewster's Encyclopedia 639 

1161-1164. Blakey's Historical Sketch of Logic 639 

1165. This Work Not a Book on Psychology 641 

1166-1176. Prince's Nature of Mind 641 

1177-1196. Haig's Symbolism 644 



PREFACE. 

This book describes a new system of logic by which reason- 
ing can be carried on by an infallible process. All the implied 
meanings of sentences and of collections of facts, can be as 
infallibly and easily determined, by this new method, as the 
interest on a promissory note can be ascertained by mathe- 
matical rules. 

The new system is based on axiomatic principles and gov- 
erned by infallible laws. Its method consists in the repeated 
use of a few processes which are performed in a mechanical 
manner, and the results appear automatically. 

It is easy to learn, and probably the time is not far distant 
when it will take the place of the syllogistic and algebraic 
systems of logic now in current use. 

As a means of discovering the truth in regard to any dis- 
puted question, whether of words or facts, it acknowledges no 
equal. 

Its tools are a few simple signs, namely, the capital letters 
of the alphabet to represent positive terms, the small letters to 
represent negative terms; the mathematical sign of equality, 
=, for "is"; a short perpendicular mark, |, for "or" and a 
square for the "universe of discourse." 

When a square is subdivided into the proper number of sec- 
tions it is called a Reasoning Frame. By the use of the Rea- 
soning Frame every proposition which can possibly be made 
with the letters used, is set before us. We then eliminate 
every proposition which is inconsistent with the given proposi- 
tion or state of facts. 

The uneliminated propositions which automatically remain 
in the Reasoning Frame will then give us every iota of truth 
which our data will yield. 

It is beyond dispute, that if we make every proposition which 
it is possible to make with the given terms and then eliminate 
the inconsistent propositions, the consistent ones must remain. 

THOMAS 1). HAW LEY. 
6107 Madison Ave., Chicago, 111. 
September 1, 1896. 



INFALLIBLE LOGIC. 

INTRODUCTION. 

1. This work is for the use of lawyers, ministers, teachers, 
students and for everyone who is interested in the art of rea- 
soning, whatever may be the general or special object he has in 
view. 

2. One of my chief objects has been to economize the time 
of the reader and therefore I have contented myself with the 
briefest and most diagrammatic account of logical facts and 
theories. 

3. So far as the work contains a description of the Reason- 
ing Frame, its methods, results and the discoveries which it 
has led me to make, it is original, but with regard to the old 
logic, I have borrowed liberally from the works of the best 
writers on that subject. 

In such a subject as logic, it is hardly possible to have any 
ideas, the germs of which are not to be found in the works of 
preceding writers. 

My acknowledgments are due in the first place to Keynes' 
"Formal Logic." I also owe much to the following works: 
Venn's "Symbolic Logic;" Jevon's "Lesons in Logic;" Miss 
Jones' "Elements of Logic;" "Studies in Logic," by members of 
Johns Hopkins' University; Bain's "Logic" and "Whately's 
"Elements of Logic." 

4. While I have the greatest respect for the authors of these 
works, still I have thought there was room for another work 
describing fully a new system of reasoning. 

5. In order to avoid elaborate descriptions, I have intro 
duced a large number of diagrams to explain the working of 
the new system. It is a merit of this system that the eye can 
see that its results are correct. Thinking is made visible. I 
have aimed to be plain, concise and intelligible. 

6. I have written for the first beginner in logic and for the 
most advanced logician. The first may think that some parts 
of the work are too obstruse; the other, that some parts are 



2 INTRODUCTION. 

too puerile, but I hope that each will find something interesting 
and profitable. 

7. It differs from all other works on logic in this ; it explains 
a graphic and diagrammatic system of logic, which, so far as 
I know, has never before been exhaustively described. 

8. The object of the system is to teach a person how to rea- 
son correctly, how to detect fallacies, how to deduce the latent 
meanings of propositions, how to draw at one operation all the 
conclusions which necessarily follow from a large number of 
premises containing a large number of terms, no matter wheth- 
er the propositions are categorical, disjunctive, hypothetical, 
affirmative or negative, simple or complex. 

9. When propositions which are contradictory to each other 
are expressed in the Reasoning Frame, it immediately reveals 
their contra dictoriness. It also enables us to perceive intuitive- 
ly what propositions in the Reasouing Frame are inconsistent 
with the premises. 

10. It makes inductive reasoning as easy and simple as 
deductive and thus enables us to tell exactly what the facts in 
any given case establish. It also enables us, given any proposi- 
tion, to find all the propositions which are equivalent or con- 
tradictory to the given proposition. 

11. If the reader will turn to the chapters on Examples and 
on Fallacies, he will get a good idea of the difficult problems 
which this system of logic is capable of solving. 

12. I advise the reader to work the examples given. You 
may understand the rules laid down but practice is necessary 
in order to remember them. You should draw every figure and 
strike out the combinations which are to be eliminated until 
you can repeat the process without looking at the book. Begin 
slowly and carefully and in a short time with practice you can 
eliminate inconsistent combinations at sight and then you will 
find that your reputation for ability to reason correctly will 
enable you to rank with the best thinkers of the day. 

I shall be happy at any time to answer any questions on the 
new Logic that may occur to my readers. I do not profess, how- 
ever, to know much about the old logic. 



CHAPTER I. 

LOGIC. 

13. Logic is the science of Interpretation. It is the art of 
interpreting, explaining, and expounding that which is not 
obvious. Its function is to make plain, clear and intelligible the 
implied meanings of propositions. It unfolds that which is hid- 
den and latent. Logic has been defined as the science of reason- 
ing, but this definition is too broafl, because it covers numerical 
reasoning, and I regard numerical reasoning as a different 
kind of reasoning. In mathematics, the axioms, the laws, the 
rules, and the problems are of a different nature from those of 
logic. Logic is not capable of solving numerical problems. 
Again the definition is too narrow, because logic is an art as 
well as a science. Miss Jones in her book, "Elements of Logic," 
defines Logic as the "Science of Propositions," but I think this 
definition is indefinite. Keynes, in his "Formal Logic," says 
that "Formal logic may be defined as the science which inves- 
tigates those regulative principles of thought that have univer- 
sal validity whatever may be the particular objects about 
which we are thinking. It is a science which is concerned with 
the form as distinguished from the matter of thought," but I 
think that the matter of thought is an important item in logic. 
The definition which I like is: Logic is the Science and Art of 
Interpretation. 

14. In reasoning, from the given proposition or proposi- 
tions, we infer all the other propositions which must be true if 
the given propositions, which are called premises, are true. In 
other words we bring to light all the latent meanings of the 
premises. Sir W. Hamilton says, "Reasoning is the showing 
explicitly that a proposition not granted or supposed, is implic- 
itly contained in something different, which is granted or sup- 
posed." In reasoning logically we show in the conclusion how 
much has been admitted in the premises. 



4 LOGIC. [Chap. 1. 

1F>. Reasoning will not discover any new facts or enable us 
to predict a coming event. No new fact can ever be discovered 
by logic, but we can obtain new meanings of old truths. New 
truths must come to us either by testimony or by observation 
through the senses. To get a new view of an old truth, to learn 
new names for old things, and to ascertain the correct names 
for ideas which are opposite to or are inconsistent with the 
ideas contained in the premises is sometimes of as great value 
as it is to learn a new fact. 

16. When a speaker puts forth a proposition, the proposi- 
tion has a prima facie meaning. This is the common meaning 
that almost everyone who understands the meanings of the 
words used in the proposition, would put upon it, but most pro- 
positions, if they are simple ones, have at least four meanings 
and if they are complex in their structure, they may have many 
more. It is a defect in syllogistic logic, or the old logic, as I 
shall usually call it, that it only attempts to give one new mean- 
ing out of the numerous ones which two premises furnish. A 
system which only gives a part of the truth when it ought to 
give the whole truth, is not much better than no system. 

17. Iq the system which I am about to describe we are 
enabled to draw with one operation every possible meaning 
which is consistent with the premises, no matter how complex 
the premises are, or how many there are of them. Of course, it 
will be understood by the reader, that before the mind can rea- 
son, it must know the meaning of the various words used in 
framing the premises. I do not doubt that animals can reason 
without language, but it would be almost impossible for 
human beings to reason with each other without employing 
language. Logic is concerned in the first place with language, 
and in the second place with the ideas which the terms used 
stand for. 

18. Intellectual thought is usually divided into sensations, 
perceptions, ideas, conceptions and notions, but I shall use the 
term ideas as a general name to stand for all these different 
kinds of thought. Ideas also represent things. Things them- 
selves are not in thought, only the ideas which represent the 



§§ 19-21. J OPPOSITE IDEAS. 5 

things perceived by the mind are in thought. An idea may be 
simple, complex, or compound, but whichever it may be, it is 
always a whole, a unit. 

19. The fundamental theory of this new system of logic is 
that every idea has its opposite, and that the two are insepa- 
rably joined together in thought. It is impossible to separate 
them, they have one common life. If you destroy one, you neces- 
sarily destroy the other, if you posit one, you necessarily posit 
the other. 

20. We may describe the idea, and its opposite as positive 
and negative. These are probably the best terms we have. I do 
not like the term negative in this connection, because it usually 
implies either negation or the absence of some quality, but the 
truth is, that a negative idea is just as positive really as a posi- 
tive idea. The terms positive and negative are purely relative 
to each other. We might call the positive, negative, and the 
negative, positive, and it would make no difference in the 
results. The fundamental thought is, that they are simply 
opposites. 

21. When a term which is used as the name of an idea has 
the prefix "not" attached to it, we will call that a negative 
term, and the idea which it represents, a negative idea. When 
the term used is without the prefix "not," we will call that a 
positive term and its idea a positive idea. The reasoning pro- 
cess, in order to be correct, must deal with a positive and its 
negative, or with a negative and its positive at the same time. 
Unless this is done, the results of the reasoning process will be 
incorrect. The whole art of the reasoning process proper con- 
sists in positing a negative term with every positive, and a pos- 
itive with every negative, and of then making every possible 
combination, or proposition, which can be made out of these pos- 
itive and negative terms, and of then selecting the combina- 
tions, or propositions, which are consistent with the premises, 
by eliminating those which are inconsistent. This must neces- 
sarily and infallibly give us every possible proposition which is 
consistent with the premises. 



6 LOGIC. [Chap. 1. 

22. It is axiomatic, that if we have two kinds of propositions, 
namely, those which are consistent with the premises, and 
those which are inconsistent, and we remove one kind, the 
other kind must remain. If I hold a number of white balls and 
a number of black balls in my hand, and take away the black 
balls, the white balls must remain. This process of combining 
the expressed and implied terms of a proposition, and of elimin- 
ating the inconsistent propositions thus obtained, is the only 
way by which the mind can learn all the true meanings which 
are hidden, or latent, in any proposition. 

23. The old logic, which is based upon the mutual relations 
of two classes, can never yield satisfactory results, because it 
is based upon a wrong theory. According to the old logic, we 
must either reason from the whole to the parts, or from the 
parts to the whole. The first was called deductive reasoning 
and the second inductive. This is an exceedingly narrow and 
imperfect view of the reasoning powers of the mind, as we 
shall show further on. 

24. A little reflection will convince the reader that the 
results which can be obtained by our combining and eliminat- 
ing system must be as accurate as those which can be obtained 
in the solution of arithmetical problems by the processes of 
addition, subtraction, multiplication and division. There is a 
pronounced analogy between our logical method and the arith- 
metical method. k With both systems it is necessary to have 
correct data. The computation may be correct, but if the items 
of an account are wrongly stated, a correct computation alone 
will not give correct results. Likewise in logic, our premises 
must be true, or our conclusions will be false. In arithmetic the 
data must be definite or the problem cannot be solved. If I were 
to ask a scholar what some apples, at some cents apiece, would 
cost, he could not tell me. And similarly in logic, if our prem- 
ises are indefinite, we can arrive at no definite conclusion. 

25. The great superiority which mathematical reasoning 
has hitherto enjoyed over logical reasoning is principally due 
to the fact that mathematical problems are usually stated in 
definite and unambiguous terms. Archbishop Whately quaint- 



§§ 26, 27.] THE THEORETICAL SCIENCES. 7 

ly says: "It is a wise remark of Dr. Barrow, that 'Confusion is 
the mother of iniquity.' " 

26. Logic is especially useful in solving the problems of the 
so called theoretical sciences, such as Jurisprudence, Law, 
Political Economy, Politics, Metaphysics, Philosophy and The- 
ology. In the past it has suffered considerably at the hands of 
its over-zealous friends who have made the most extravagant 
claims for its usefulness in every department of knowledge. 

27. It is worth while to remark here that logic has little to 
do with Grammar and Grammar has still less to do with logic. 
In pursuing its own methods, logic must be independent of 
Grammar, and if it is necessary for us to violate the rules of 
Grammar, in order to obtain logical precision, we should do so 
without any hesitancy. In order to reason logically, we must 
use logical terms, logical signs, and logical laws. Grammar 
must take care of itself, and it must allow logic to do the same 
for itself. 



CHAPTER II. 

DEFINITIONS. 

28. Logic being a technical science, the terms used are 
not always employed in their common acceptation, and it is 
necessary, therefore, in order to guard against mistaken mean- 
ings, that we should define the terms we employ. And when we 
have once put a definition upon a term we should not change 
it, but adhere strictly to the meaning given. When no defini- 
tion is given for a word it will be understood that the word is 
used in its popular and customary sense. When we define a 
word we usually state in a single word, or a phrase, the particu- 
lar circumstance of resemblance or of difference which the 
word has to other objects. 

29. Definitions are divided into nominal and real. A nom- 
inal definition is used to explain the meaning of the word used. 
For instance, the meaning of the word decalogue would be, the 
ten commandments. A real definition is used to explain the 
nature of the object, as for instance, silver can be defined as a 
white, heavy, fusible metal, used largely to make coined money. 
In the exact sciences, where words are used strictly, the nom- 
inal and real definitions are usually the same. For instance, the 
definition of a circle or of a triangle would include the quali- 
ties which are implied by the name. 

30. Accidental definitions enumerate properties which can 
be considered as belonging accidentally, but not necessarily, to 
the word, as for instance, George Washington was born in Vir- 
ginia, was the first President and was called the Father of his 
country. 

31. DefinitioDs should neither be too broad nor too narrow. 
They should be full enough to convey a clear meaning of the 
word. And, again, a definition should be plainer and clearer 
than the word which is defined, otherwise the definition would 
be useless. It should be brief, and yet not so brief as to be 



§§32-35.] LOGICAL DIVISION. 9 

obscure, nor should it be so long as to become prolix. Tauto- 
logical definitions are not elegant so far as their phraseology is 
concerned, but they are most accurate in their nature, and 
when one wishes to be exceedingly precise in his definitions 
a tautological definition will be the most exact one that he 
can use. 

32. Logical definition grows out of logical division. In a 
logical division we divide the genus into its different species 
and the species into the different individuals; thus animal 
would be a genus; horses, dogs, etc., would be species, and then 
we could divide the species into the parts or the individuals 
which compose the species. 

33. Physical division differs from logical division in this: 
That physical division separates an object into different parts. 
Thus, a man could be divided into head, body, limbs, organs, 
etc. In logical division we should divide the whole so that each 
part would be less than the thing divided, and all the parts 
into which the thing is divided should exactly equal the whole, 
and none of the members should be contained in one another. 
In other words, we should not be guilty of the fault which is 
called cross-division. For instance, if we divided books into 
English, French, quarto, octavo, etc., we should violate the 
rules of logical division. 

34. Classification differs from division in this respect: That 
when we want to classify objects we separate them according 
to some difference, and we continue the separating process until 
a difference is no longer wanted, or until it cannot be found. 
Cuvier's system of classification was as follows: Individual, 
species, kind, family, order, class. 

35. I like Hobbe's definition of the word name. He said, 
"A name is a word or set of words serving as a mark to raise 
in our minds a given idea, and also to indicate to others what 
idea is before the mind of the speaker." In this work I shall 
use the word name in a general sense to cover appellations, 
patronymics, titles, designations and descriptions. So that a 
name may be a single word or it may include a number of 
words. For instance, when I say "Grover Cleveland is the 



10 DEFINITIONS. [ Chap. 2. 

President of the United States," the President of the United 
States is another name for the same person. "The British 
Museum is the largest collection of books in the world in one 
building." In this sentence, the largest co^ection of books in 
the world in one building is a name for the same thing that 
the British Museum is a name for. 

36. Adjectives are names, thus: "George is wise;" wise is a 
name for George. Adjectival names are almost always indefi- 
nite, but a sentence which is indefinite, because the second 
member of it is an adjective, can be made definite by repeating 
the subject of the sentence after the adjective. Thus the indefi- 
nite sentence, "this man is wise" becomes definite in "this man 
is this wise man." This sentence read backward, "this wise 
man is this man," is equally definite. 

37. A definite sentence will have the same meaning when 
read backward that it has when read forward. In logic, which 
should be an exact science, indefinite propositions should 
always be converted into definite ones, by repeating the subject 
in the predicate, or conversely before we proceed to reason 
from them. 

38. A name, it seems to me, always indicates the existence 
of the idea or thing spoken of. Existence does not necessarily 
mean material existence. It may mean ideal or mythical exist- 
ence, but it must mean some kind of existence. 

39. There is a difference between singular and general 
names. Singular names apply to individuals; New York, the 
Alleghanies, the British Museum, designate each a particular 
thing, but general names apply to objects which belong to a 
class; man, animal, book, house, are examples of general 
names. General names are said to be connotative because 
they imply that the members of the class have certain attri- 
butes in common. 

40. A general name, like an adjectival name, is always 
indefinite, and when the predicate of a sentence is a general 
name, before it can be treated logically it must be made defi- 
nite; thus "man is an animal" is an indefinite sentence, because 
when it is read backward, "an animal is a man," we know as a 



§§4143.] GENERAL TERMS. 11 

matter of fact that it is not true. But we can convert it into a 
definite sentence by making it tautological; thus, ''man is an 
animal man." This is a true sentence when read forward or 
backward. 

41. One of the mistakes of the old logic is that it attempts 
to reason with general terms without rendering them definite. 
According to the old logic 'mien are mortal beings" would be a 
good logical proposition, but a little examination will show us 
that it is not a definite proposition, because if it were definite 
it would be just as true a proposition, when read backward, as 
when read forward. But when we read it backward, "mortal 
beings are men," we know, as a matter of fact, that many mor- 
tal beings, such as horses, dogs, etc., are not men. But the 
sentence, "men are mortal beings," can be made into a definite 
and true sentence; thus, "men are mortal men." It is now in 
the proper shape for an exact logic to deal with. One of the 
principal safeguards against drawing false conclusions is the 
conversion of indefinite propositions into definite ones by 
repeating the subject in the predicate or conversely. Accord- 
ing to Sir William Hamilton, a general name is a concept, but 
I prefer to use the term concept to represent abstract terms, 
such as justice, whiteness, etc. 

42. There is a class of singular names called individual 
names. Individual names are non-connotative; they serve to 
mark a particular person or thing without implying any qual- 
ities. Thomas, George, John, etc., are examples. 

43. A name is called a term when it is used as the subject 
or predicate of a sentence. A term is also called a predicable. 
There are five kinds of predicables in the old logic, viz: Genus, 
species, difference, property, accident. In the sentence, "man 
is an animal," man is the species and animal is the genus. In 
"man is a rational animal," rational is the difference. In "this 
man is tall and ignorant," tall and ignorant are accidents. In 
"the magnet has polarity," polarity is a property. The same 
word may be used to exhibit each of the five classes of predi- 
cables; thus, red in relation to a gown is an accident, to blood 



12 DEFINITIONS. [ Cbap. 2. 

it is a property, to a rose it is a difference, to pink it is a 
genus, to color it is a species. 

44. Another class of names is called relatives. A relative 
name always implies a correlative. Parent and child, husband 
and wife, father and mother are examples. Positive and nega- 
tive also express relativity. We might call parent positive 
and child negative, or vice versa. Logically, it would make no 
difference. We could let A stand for either one and not-A 
stand for the other. It must always be borne in mind that in 
logic the term "negative" does not necessarily mean negation 
or the absence of a quality. It does, however, imply a differ- 
ence. When I say, "A horse is not a giraffe," I mean, that a 
horse is different from a giraffe. "Not a giraffe" is a name or 
description applied to a horse. If I should say "a giraffe is not 
a horse," then, "not a horse" would be a name applied to a 
giraffe. 

45. Negative names, if indefinite, when used as predicates, 
must be converted, in the manner heretofore pointed out, into 
definite names before they are proper subjects for logical 
treatment. Some logicians call negative terms infinite terms. 

46. A word which can be used by itself for a name is called 
a categorematic word and a w r ord which must be used in con- 
nection with others to make a name is called a syncategore- 
matic word, e. g., prepositions. 

47. A name which can be applied to a single whole that 
is made up of a number of individuals is called a collective 
name. Army, congress, supreme court, are examples. 

48. When the mind perceives an abstract idea, it is said to 
apprehend it. This act of the mind is very similar to the act 
of the mind when it perceives a material object. The old logic 
says that judgment is the act of comparing the perceptions or 
apprehensions of the mind and of deciding whether they agree 
or disagree in their qualities. I think this is a mistaken idea. 
I believe that a judgment is the act of the mind in giving a 
name to one of its ideas in addition to the name by which the 
idea is commonly known. But before we can determine what 



§§ 49-54.] JUDGMENTS. 13 

a judgment is we must define the word "is." When I say "this 
thing is salt," I mean "this thing has the name of salt;'' 
when I say, "this thing is chloride of sodium," I 
mean that "this thing has the name of chloride of sodium." 
So we see that "is" logically means "has the name of." Now 
when I say that, "salt is chloride of sodium," I express a judg- 
ment of the mind. The sentence, "salt is chloride of sodium." 
means that the same identical thing, has two different names, 
viz., salt and chloride of sodium. So that a judgment is the 
giving of two names to one thing. 

49. A judgment expressed in words becomes a proposition, 
and a series of propositions becomes a discourse. A discourse 
can be analyzed into propositions, and propositions can be 
analyzed into names, so that in a sense the science of logic is 
the science of names. The complete interpretation of all the 
implied meanings of names used in propositions is the art of 
logical reasoning. 

50. A term is a word or any combination of words used as a 
name for a subject or a predicate of a sentence. A single word 
used as a name is called a simple term. 

51. Where two or more names are joined together to make 
a subject or a predicate we have a complex term. In the sen- 
tence, A is B or C, B or C is a complex term. This kind of a 
proposition is called a disjunctive proposition. The elements 
of a disjunctive term are called alternates. 

52. Positive and negative terms, such as A and not-A are 
called opposites in our system, because the same thing cannot 
have the name of A and the name of not-A at the same time. 

53. Terms which are incompatible with each other are 
called contraries; thus, white and black are contraries, but 
white and not-white are opposites. Another name for oppo- 
sites is inconsistents. This means that they cannot stand 
together, that is, that they cannot both be applied to the same 
thing. 

54. The old logic divides propositions into universal and 
particular. When the predication is made of the whole of a 



14 DEFINITIONS. [ Chap. 2. 

subject it is universal, and when it is made of a part of a subject 
it is particular. This is called the doctrine of quantity. This 
division, it seems to me, is not a sound one. The subject of a 
proposition must be an idea, and that idea must be a whole, a 
unit. If we subdivide an idea into its parts then each part 
becomes a whole or unit idea. Whenever the mind takes an 
idea for a subject it must take it as a whole. There is no such 
thing as taking a part of an idea, unless that part is taken as 
a whole. It makes no difference how far we may carry the 
process of dividing and subdividing an idea, whatever part we 
may take we must speak of the whole of that part. The idea 
that the mind can-apply a predicate to a part of an idea is one 
of the strangest misconceptions of the old logic and one which 
is fatal to any correct logical system. 

55. The old logic also divides propositions into affirmative 
and negative. Affirmative means that the predicate is affirmed 
of the subject, and negative means in the old logic that the 
predicate is denied of the subject. This is called the doctrine 
of quality. It seems to me this is another error. The copula 
of a proposition is, always, is; the negative particle "not" does 
not belong to the copula, but it belongs to the predicate; it is 
a part of the name of the predicate. A proposition, no matter 
whether it contains the word <k not" or does not contain it, is 
always affirmative; it affirms that the subject and predicate 
are two names for one thing. Now names may be affirmative 
or they may be negative, but the proposition is always affirma- 
tive. 

56. The doctrine of quantity and the doctrine of quality has 
given rise in the old logic to four forms of propositions. The 
first form is called the universal affirmative, and its formula is 
all S is P. In the formula S stands for subject and P stands for 
predicate. It means all S is some P. In this form it is an 
indefinite proposition, and before using it it should be changed 
to the form of all S is PS. This makes a logical proposition 
out of it, and we can read it backward, all PS is S, which is 
equally true with the proposition that all S is PS. 



§3 57-60.] THE PARTICULAR AFFIRMATIVE. 15 

57. The second form is called the particular affirmative, and 
its formula is some S is P, meaning some S's are some l v s. It 
is said by the old logic that the subject some S means that the 
subject is not taken altogether, and that the word some is 
indefinite. Now it may be true that in grammar the word some 
is called an indefinite adjective, but because it is indefinite in 
grammar that does not make it indefinite in logic; the two 
sciences are distinct and must be kept distinct. When I say 
''some people are patriotic," I am not taking a part of my sub- 
ject, — my subject is "some people," and I am taking the whole 
of my subject. I mean by the word "some people," all of the 
people that I have in my mind. True, I may not know how 
many of them there are, but that is a different question. When 
I say "the American people are patriotic" I do not know how 
many Americans there are in this case any more than I know 
how many people there are when I say "some people," but no one 
will pretend that "the American people" is an indefinite sub- 
ject; and I contend that "some people" is not indefinite in logic 
any more than the phrase "the American people" is 
indefinite. When I say "some people" are patriotic it is per- 
fectly clear that by the word "some" I mean "patriotic " people, 
so that the proposition means patriotic people are patriotic. In 
this last case the term "patriotic" being an adjective is indefi- 
nite; it can be converted into a definite predicate by combining 
the subject with it, thus giving us the tautologous proposition 
patriotic people are patriotic people, and nothing can be truer 
than that is. 

58. The third form is called the universal negative, and its 
formula is No S is P, or properly, No S's are P's. Its logical 
effect is to affirm that the name B goes with the name Not-P. 

59. The fourth form is called the particular negative, and 
the formula is "Some S is not P." As I have said before, "some 
S" means the whole of the class which we have in our mind. 

CO. These four forms have four symbols, A, I, E, (), which 
are taken from the Latin words afflrmo and ncyo; affinno means 
I affirm, and nego means I deny. A and I are the first two vow- 
els of the word affirmo, and stand respectively for the universal 



16 DEFINITIONS. [Chap. 2. 

affirmative and the particular affirmative, and E and O are the 
vowels inncgo and stand respectively for the universal negative 
and the particular negative. The old logic is based on these 
four forms of propositions. Yet we find on examination that 
the distinction between universal propositions and particular 
propositions has no basis in logic; in reasoning we must take 
the whole of our subject. Neither do we find that the division 
of propositions into affirmative and negative has any sound 
basis. Every proposition is affirmative and must be so in the 
very nature of things. This view will be more fully developed 
later on. 

61. Some subjects are called Indesignate, for instance: 
Comets are subject to the law of gravitation. Dr. Keynes says 
that indesignates are generally taken as universal. This is 
correct when we know as a matter of fact that all comets, or 
whatever the subject may be, are intended. But, unless we 
are perfectly sure that an indesignate proposition is intended 
to be taken universally we should throw it into the proper 
logical form so that there can be no doubt about the matter. 

62. Another theory of the old logic is called the division of 
terms into distributed and undistributed. This division is 
arbitrary and technical. The word distributed corresponds 
to the word universal as applied to propositions, and the word 
undistributed corresponds to the word particular. Distributed 
means that the subject or the predicate is undivided, that is, 
that it is taken as a whole. It implies that the subject or predi- 
cate which is said to be distributed, is a definite term. 

63. The word undistributed implies that a part of the sub- 
ject or the predicate is taken and that the term is indefinite. 
An undistributed term usually has as a part of its name one of 
the words "few, some, many or most." These words are said 
in the old logic to indicate that only a part of the term is taken. 

64. This division of terms into distributed and undistrib- 
uted has, it seems to me, no more foundation in logic than the 
division of propositions into universal and particular has. 

65. We have seen that the subject and the predicate of pro- 
positions are two names for the same thing. Now an idea with 



§§ 66, 67.] INDEFINITE PROPOSITIONS. 17 

an indefinite name, like a wheelbarrow with only one handle, 
is not good for much, and an idea with two indefinite names is 
as useless in logic as a wheelbarrow without any handles; it is 
just as bad to have an indefinite predicate as it is to have an 
indefinite subject. The old logic places a good deal of impor- 
tance on the meanings of the terms "subject" and "predicate." 
But, as we have seen, the subject is a name, and the predicate 
is a name and the only real difference between them is that one 
comes first and the other comes last; both being names of the 
same thing, it is a matter of no importance, from a logical 
standpoint, which comes first. We may say indifferently salt 
is chloride of sodium or chloride of sodium is salt; both mean 
the same thing, and in logic one form is as good as the other. 
In a proposition the subject affirms that it is the name of an 
idea ; the predicate does the same. Strictly speaking the predi- 
cate is not a predication about the subject; it is a predication 
of the idea which it represents. The subject likewise by its 
very existence predicates that it is the name of the same idea 
for which the predicate is a name. 

66. In the universal affirmative proposition A, the subject, 
only is said to be distributed; the predicate is indefinite and 
therefore undistributed. An example, "All men are mortal," 
shows this. The term "mortal" is an adjective and indefinite. 
It applies to many other kinds of beings than men; it is there- 
fore not a definite name for men. The proposition, therefore, 
is not strictly true. If it were strictly true we could say "All 
mortals are men," but this we cannot do because we know as a 
matter of fact that horses, dogs, etc., are mortal and yet they 
are not men. 

67. A logical proposition must have a logical form. When 
it is in a logical form it will say just what it means. A model 
form of a proposition is "I am I." "Salt is salt," "A quintillion 
is a quintillion." The mind intuitively perceives the truth of 
these propositions. I may not know what a quintillion is, but 
still I know that a quintillion is a quintillion. Every true 
proposition can ultimately be reduced to the formula "A is A," 

2 



18 DEFINITIONS. [Chap. 2. 

and every false proposition to the formula "A is not- A." And 
hereby we can solve the old problem of "What is truth?" Truth 
is the giving of the right name to the right thing. 

68. The universal affirmative can be easily converted into a 
logical proposition by repeating the subject after the predicate, 
thus: "All men are mortal" becomes logically "All men are 
mortal men." The mind immediately perceives the truth of 
this proposition. But in the form in which the old logic puts 
it we must reject it as being indefinite and unfit for logical 
treatment. 

69. The particular affirmative proposition I does not distrib- 
ute the subject. An example is: "Some men are tall," and we 
know that this means "Some men are some tall beings." Both 
terms are indefinite and unless we can reform the proposition 
and make both terms definite we shall have to reject it. But 
we can reform it. When reformed it becomes "Tall men are 
tall men." This is a true proposition. 

70. The universal negative proposition E distributes both 
subject and predicate. An example is "No men are immortal." 

71. The particular negative O distributes the predicate, but 
not the subject. "Some men are not tall" is an example. 
When we ask who the "some men" designated by the subject 
are we know that the "some men" are the same men who are 
called in the predicate "not tall," so that our subject really 
means "Men who are not tall," and when we ask what does the 
word "tall" in the predicate refer to, the answer must be that 
it refers to the same men described by the subject. So that the 
proposition really means "Men who are not tall are not-tall 
men." In this form it is strictly true and can be read back- 
ward or forward indifferently . 

72. Dr. Keynes in his work on Formal Logic divides propo- 
sitions into several other classes. He says "Copulative propo- 
sitions are complex propositions which can be analyzed into a 
conjunction of two or more affirmative propositions having the 
same subject, and remotive propositions are those which can be 
analyzed into two or more negative propositions having the 
same subject." 



§§74-79.] KINDS OF PROPOSITIONS. 19 

73. Exceptive propositions are propositions in which the 
subject is limited by some such word as 'unless' or 'except/ 

74. "Exclusive propositions contain some such word as 
'only' or 'alone/ whereby the predicate is limited to the 
"subject.' " 

75. It is sometimes difficult to translate into a logical form 
the words "unless" and "except," especially when the proposi- 
tion which contains those words is taken out of its connection. 
The meaning of a proposition is frequently contained in the 
proposition and its context taken together. By taking a propo- 
sition with its context we can almost always determine the 
logical force of the proposition. If I say "I am going to town 
unless it rains," that may mean I am going to town if it does 
not rain, or if it rains I am not going to town, or I am going to 
town or it will rain. The word "except" usually has the mean- 
ing of "not." If I say "A is 15 except when it is C," that may 
mean A is B when A is not C, or it may mean A is B when B is 
notC. 

70. In common speech language is generally elliptical and 
indefinite. We seem to be unwilling to take the time or trouble 
to express ourselves with precision. We are contented with 
giving in as brief terms as possible a general idea of our 
thoughts. But when it becomes necessary to reason, then our 
thoughts and our language must be clear, definite and exact. 

77. Propositions are also divided into real, verbal, and for- 
mal. A proposition which contains more information than a 
mere name would give is said to be real. For instance, "The 
angles of any triangle are together equal to three angles." 

78. A verbal proposition gives no information except that 
which is contained in the meaning of the term. Examples are: 
"Salt is chloride of sodium;" "Charles Egbert Craddock is Miss 
Murfrey." 

79. A formal proposition is one in which signs are used to 
stand for the terms, and the proposition intuitively appears to 
be true. Examples are : "A is A ;" "All A is either B or not B." 
This example has been given as a formal proposition: "If all A 
is B and all B is C, then all A is C." But this example is not 



20 DEFINITIONS. [ Chap. 2. 

satisfactory. Suppose we substitute a concrete example con- 
structed in a parallel manner and test it: "If this rose is red, 
and red is a color, then this rose is a color;" "If this finger ring- 
is gold, and gold is heavy, then this finger ring is heavy;" "If 
this man is one, and one is a number, then this man is a num- 
ber." These arguments are not logical. But if we make the 
premises definite we shall be saved from the error of drawing 
conclusions which are not warranted by the premises. Thus 
if we say, "This rose is a red rose" we will not take the next 
step and say "A red rose is a color." If we say "This finger ring- 
is a gold finger ring" we will pause before we say "This gold 
finger ring is heavy." If we say "This man is one man" we are 
not likely to say that one man is a number. 

80. Real propositions, again, are divided by the old logic 
into true and false. This is another of the useless refinements 
of the old system. It is extra-logical. Logic has nothing to do 
with the truth or falsity of propositions. The problem of logic 
is, given any proposition, what are all the implied meanings 
of that proposition? The other sciences must determine 
whether propositions do or do not contain matters of fact. 

81. When different objects resemble one another in some 
particular, though differing from each other in a hundred other 
particulars, the mind can perceive this partial resemblance of 
objects, and give a general name to the resemblance. This is 
called generalization. For instance, some animals are called 
quadrupeds because they agree in having four legs. The 
objects to which a general name is applied are called a class, 
and the objects are said to be contained in the class. It seems 
to me that this is an erroneous idea. When I try to realize in 
my mind the idea of a horse being contained in a (fuadruped, I 
find it impossible to do so. How is it contained in a quadru- 
ped? The truth is that the quality of being a quadruped is in 
the horse. When I say "Man is an animal," what I really 
mean is that the quality of animality is in a man, and not the 
ridiculous notion that a man is in an animal. 

82. These general terms logically have the nature of adjec- 
tives. When I say "George is wise" I do not mean that George 



§§ 83-87.] NAMES. 21 

is contained in wise, but I mean that the quality of being wise 
is in George. Consequently whenever we meet with a propo- 
sition which ends with a general term we must treat it as an 
adjective and make the proposition definite by adding the sub- 
ject to the predicate, and then our propositions will be definite 
and true. Thus "A horse is a quadruped-horse;" "Man is an 
animal-man;" "George is wise George." When we do this, 
it will be impossible to draw any false conclusions from a pre- 
mise rendered definite in this way. 

83. Abstract names, such as "whiteness," "benevolence," 
"kindness," "friendship," are a result of the generalizing pro- 
cess. They are names of qualities without containing any ref- 
erence to the objects which possess the qualities. 

84. The names of things, that is material things, are called 
concrete names. The distinction seems to be that an abstract 
name is the name of a quality, and a concrete name is the name 
of a thing possessing qualities. This is extra-logical, and a con- 
crete name may be used as an abstract name, and conversely. 

85. When we consider a name in regard to the objects to 
which it is applied, we consider its extension. When we con- 
sider it with regard to its qualities we consider its intension. 
This also is an extra-logical distinction. 

86. According to J. S. Mill, names which are considered with 
regard to their extension and intension are connotative. 
According to Sir Wm. Hamilton, denotation corresponds with 
extension, and connotation with intension. 

87. Kant subdivided propositions into four divisions, and 
each division into three subdivisions. The divisions are, Quan- 
tity, Quality, Relation and Modality. Quantity is divided into 
singular, "This is a horse;" particular, "Some horses are black; " 
universal, "All horses are quadrupeds." Quality is divided 
into affirmative, "All men are mortal;" negative, "No man is 
immortal; infinite, all men are not immortal." Relation is 
divided into categorical, "Man is mortal;" hypothetical, k 'If it 
does not rain I shall go to town;" disjunctive, "Washington 
was born in Virginia or Pennsylvania." Modality is divided 
into problematic, "John may be rich;" assertoric, "John is 
rich;" appodeictic, "Body must have weight." 



22 DEFINITIONS. [ Chap. 2. 

88. Very few of these fine-drawn distinctions are of any log- 
ical importance. In regard to quantity in a true logic, all 
propositions must be universal in both subject and predicate, 
in other words, the proposition must be definite. In regard to 
quality all propositions must affirm something, that is, be 
affirmative, or they would not be propositions. A negative 
name does not make a negative proposition. In relation to 
modality the divisions into problematic, assertoric and appo- 
deictic are of no use at all in logic. The divisions of relation, 
however, are important. A categorical proposition asserts 
without any condition that a certain thing or idea has two 
names. But a hypothetical proposition indicates a certain 
amount of doubt as to whether a name given to an idea is the 
correct name or not. "If" is the word which usually implies 
the doubt. A hypothetical proposition, though, can be con- 
verted into a categorical proposition by the use of the words 
"the case of." Take the proposition "If Caesar was a tyrant 
he deserved death," and we can say "The case of Caesar being a 
tyrant is a case of Caesar deserving death," this is a categorical 
proposition. Very frequently we can treat a hypothetical 
proposition as a categorical proposition by supplying the 
implied premise. Thus, Caesar was a tyrant; tyrants deserve 
death (the implied premise), therefore Caesar deserved death." 
Nearly all hypothetical arguments are cases of a categorical 
argument with a suppressed or implied premise. When we 
supply the-implied premise there is no difficulty in dealing with 
hypothetical propositions. 

89. In a disjunctive proposition one or both of the terms has 
two names which are separated by the word "or." There has 
been a great deal of discussion by the old logicians as to the 
logical force of the word "or," some contending that "or" is not 
exclusive in its meaning; that it may mean one or the other or 
both; others contending that it is exclusive and means one or 
the other and not both. I contend that "or" should be exclu- 
sive, and at the same time it may mean both when that meaning 
is expressed. By "exclusive" I mean that of the three forms 
one or the other or both, it can only have one of them; it cannot 



§ 89.] OR. 23 

have two. Ordinarily there is no doubt that it means one or 
the other and not both. "Washington was born in Virginia or 
Pennsylvania;" one or the other can be true, but not both. 
"Columbus discovered America in 1492 or 1592." One or the 
other may be true, but not both. In this sentence, "A gem is 
rare or beautiful or both," either one of the alternatives may be 
true. But if it was not expressly stated that a gem was both 
rare and beautiful, it seems to me that the fair implication of 
the words "rare or beautiful" is that it means one and not the 
other. In this system we shall always construe "or" to mean 
one or the other and not both, unless it is expressly stated or 
manifestly implied in the context that it may mean both. 



CHAPTER III. 

THE LAWS OF THOUGHT. 

90. The question which we are now to consider is this : Can 
there be an infallible logic or system of reasoning? If we can 
find certain indisputable facts on which to base our system and 
then can find certain infallible laws by which its operations 
shall always be conducted, then it seems to me we can have an 
infallible logic. Are there such facts and such laws? 

91. The first fact is this: That all knowledge is a knowl- 
edge of differences between things. If there were no differ- 
ences between things there could be no knowledge of them. 

When I make a mark like this I could not perceive the 

mark unless it differed from the paper on which it is made. If 
the mark were exactly like the paper I could not perceive it. 
If I look out of the window at a tree the necessary con- 
dition for my perception of the tree is that it shall differ from 
other things. If a person prick me and I could not perceive 
any difference in my feelngs I should have no knowledge of the 
fact that I had been pricked. A universal sensation of same- 
ness is the same as no sensation at all. This doctrine is called 
the Law of Relativity. It lies at the foundation of all thought 
and all knowledge. Its truth is beyond dispute. The mind 
intuitively perceives that it must be true. 

92. The next fact to which we shall call attention is this: 
That when, for instance, a line is perceived by the mind, the 
mind must perceive whether the line has the quality of straight- 
ness or not-straightness. The mind cannot think of straight- 
ness without thinking of not-straightness. An object cannot 
be thought of as being high without other objects are thought 
of as being not-high. A thing cannot be thought of as being 
dead without other objects being thought of as being not-dead. 
In other words, every name necessarily implies that it has 
an opposite and that when one of these is posited in thought 



§3 93, 94.] LAW OF IDENTITY. 25 

the other must also be posited. They cannot be separated 
in thought. If the mind has one it must have the other. 
This doctrine I propose to call the Law of Opposites. This 
doctrine is a necessary corollary of the doctrine of Rela- 
tivity. The doctrine of Relativity says there must be a differ- 
ence in order to have knowledge, and the doctrine of Opposites 
says that we must have two names to express differences. 

93. The third law is called the Law of Identity. As usually 
expressed, "whatever is, is," it means that a thing is equal to 
itself. In logic it means that whatever name a thing has, it 
has that name. "John is John," means that if John has the 
name of John, he has the name of John. This is so plain that 
no amount of illustration can make it any plainer. It is an 
intuitive truth. The mind immediately perceives that it must 
be true. It is a fundamental truth. All truths cannot be 
proved, or else there would never be any end to demonstration. 
When in a course of reasoning we arrive at intuitive truths we 
can go no further. 

94. The formula for the Law of Identity is "A is A." A, 
the subject, stands for the name of some thing. A, the predi- 
cate, stands for the name of the same thing. Both subject and 
predicate refer to the same identity. When we say "A is A" 
there is only one thing that the mind has in view. Now if by 
observation or testimony the mind learns that the name "B" is 
equivalent to the name "A," then the mind can substitute B for 
one of the A's and say "A is B," or "B is A." Both propositions 
have exactly the same meaning. If the mind is given the two 
terms A and B and it has no further information respecting 
them than the fact that they are names, it cannot tell whether 
A is B or not-B. It knows that it must be one or the other, but 
that is the extent of its power. The case of "A is A" is differ- 
ent; its truth is intuitively perceived. If the proposition "A 
is B" is given to the mind and it is a true proposition, then B 
must be equal to A, and then the mind can substitute A for B 
and the proposition is reduced to the identical proposition "A 
is A," and it is impossible for the mind to doubt the truth of "A 
is A." 



26 THE LAWS OF THOUGHT. [ Chap. 3. 

95. We must either suppose that the Law of Identity is cer- 
tain, or else that there is no certainty whatever. And if a pro- 
position can be reduced to this form, everyone must admit 
either that the proposition is true, or else that the Law of 
Identity is false. 

[)&. The next law is called the Law of Contradiction. Usu- 
ally it means a thing cannot both be and not be. In logic it 
means that a thing cannot have a name and at the same time 
not have it. For instance, a line cannot, at the same time, 
have the name of straight and not-straight. A body cannot 
have, at the same time, the name of living and not-living. An 
object cannot, at the same time, have the name of high and not- 
high. It cannot be here and not-here. In other words, incon- 
sistent, or opposite, names cannot be applied to the same 
object. Consistency demands that when we give a name to an 
object we must stand by that name. We cannot say that a line 
is straight and that it is not-straight; any such course would be 
dishonest. Of course different parts of an object may have 
different qualities, and if we are speaking of the parts then 
we may say that one part is straight, of a line, for instance, 
and another part is not straight. One part of an object may 
be white and another part may be black. What the Law of 
Contradiction affirms is that the same part, the same thing, 
cannot at the same time and place have contradictory names. 

The formula for the Law of Contradiction is "Aa = 0." 
It means the non-existence of an object said to possess oppo- 
site qualities. A capital letter and its small letter are oppo- 
sites in this work. Hence every proposition which affirms the 
presence of opposite qualities is necessarily false. "AB and 
Ab" stand for opposite propositions. "ABC and ABc" are 
inconsistent, and, in fact, any two propositions, no matter how 
many terms they contain, if there is in one of them a term 
which is opposite to a term in the other, is inconsistent with 
that other and one of the propositions must be false. Thus 
"ABCDEFand ABCdEF" are inconsistent. One affirms D and 
the other affirms d, i. e., not-D, and both cannot be true. 



§§ 97-100.] LAW OF THE EXCLUDED MIDDLE. 27 

97. The next law is called the Law of the Excluded Middle. 
The Law of the Excluded Middle means, given any object, we 
can say that it either does or does not possess any given 
quality. Thus: "Sugar is sweet or it is not sweet;" "A man 
is living or he is not living; 7 ' "A dog is here or he is not here." 
It means that there are only two logical alternatives; that 
there is no middle between having and not-having, being and 
not-being. Hence the arbitrary designation, Law of the 
Excluded Middle. 

98. In logic it means that a thing must either have a given 
name or else have its opposite, that is, it must have the name 
of "sweet or not-sweet," "living or not-living," and so on. The 
formula for the Law of the Excluded Middle is "A is B or it is 
not-B." For the term "not-B" let us substitute the smaller 
letter b; this will be briefer, more convenient and quite as 
explicit. So now the formula will read "A is B or A is 
b," and hereafter capital letters will stand for positive terms 
and small letters will stand for negative terms. The small 
letters a, b, c, etc., are to be read and pronounced "not- A," "not- 
B," "not-C," etc. 

99. The Law of the Excluded Middle is a corollary of the 
Law of Contradiction. The Law of Contradiction affirms that 
of two contradictory propositions both cannot be true, and the 
Law of the Excluded Middle, that of two contradictory propo- 
sitions both canot be false. 

100. Another law is called the Law of Logical Division. It 
affirms that anything or any class can be divided into two parts 
which shall be mutually exclusive, and which, taken col- 
lectively, shall be equal to the whole. Thus we can divide A 
into B and b, and B into C and c, and b into C and c, etc. To 
take a concrete example: Bodies can be divided into living 
bodies and not-living bodies. Living bodies can be divided into 
animals and not-animals. Not-living bodies can be divided into 
minerals and not-minerals. Animals can be divided into men 
and not-men. A man can be divided into head and not-head; 
head can be divided into face and not-face. Face can be divided 



28 THE LAWS OF THOUGHT. [Chap. 3. 

into cheeks and not-cheeks, etc. This process is called 
dichotomy. The two divisions into which a thing is divided 
are together equal to the whole thing and at the same time each 
excludes the other. 



§ id.] 



AN ILLUSTRATION OF DICHOTOMY. 



29 





B 


[ c 


A 










_^ 




^^t> 


c 










B 








L_i 


a 










rH 




b 





c 



c 
c 



30 THE LAWS OF THOUGHT. [Chap. 3. 

DICHOTOMOUS TABLES. 



A B 


ABC 


A B C D 


A B C D E 


A B C D E F 


A b 


A B c 


A B C d 


A B C D e 


A B C D E f 


a B 


A b C 


A B c D 


A B C d E 


A B C D e F 


a b 


Abe 


A B c d 


A B C d e 


A B C D e f 




a B C 


A b C D 


A B 


c D E 


A B C d E F 




a B c 


A b C d 


A B 


c D e 


A B C d E f 




a b C 


A b c D 


A B 


c d E 


A B C d e F 




a b c 


Abed 


A B 


c d e 


A B C d e f 






a B C D 


A b 


C D E 


A B c D E F 






a B C d 


A b 


C D e 


A B c D E f 






a B c D 


A b 


C d E 


A B c D e F 






a B c d 


A b 


C d e 


A B c D e f 






a b C D 


A b 


c D E 


A B c d E F 






a b C d 


A b 


c D e 


A B c d E f 






a b c D 


A b 


c d E 


A B c d e F 






abed 


A b 


c d e 


A B c d e f 








a B C D E 


A b C D E F 








a B C D e 


A b C D E f 








a B C d E 


A b C D e F 








a B 


C d e 


A b C D e f 








a B 


c D E 


A b C d E F 








a B 


c D e 


A b C d E f 








a B 


c d E 


A b C d e F 








a B 


c d e 


A b C d e f 








a b 


C D E 


A b c D E F 








a b 


C D e 


A b c D E f 








a b 


C d E 


A b c D e F 








a b 


C d e 


A b c D e f 








a b 


c D E 


A b c d E F 








a b 


c D e 


A b c d E f 








a b 


c d E 


A b c d e F 








a b 


c d e 


A b c d e f 
a B C D E F 
a B C D E f 
a B C D e F 
a B C D e f 
a B C d E F 
a B C d E f 
a B C d e F 
a B C d e f 
a B c D E F 
a B c D E f 
a B c D e F 
a B c D e f 
a B c d E F 
a B c d E f 
a B c d e F 
a B c d e f 
a b C D E F 
a b C D E f 
a b C D e F 
a b C D e f 
a b C d E F 
a b C d E f 
a b C d e F 
a b C d e f 
a b c D E F 
a b c D E f 
a b c D e F 
a b c D e f 
a b c d E F 
a b c d E f 
a b c d e F 
a b c d e f 



§§ 102-104.] LAW OF ELIMINATION. 31 

102. Another law, which I propose to call the Law of Elimi- 
nation, means that if we have two different kinds of things and 
we remove one kind the other will remain. This also is axio- 
matic. If I have two kinds of numbers before me, viz.: odd and 
even, and I remove all of the even numbers, the odd numbers 
A'ill remain. If from a flock composed of white and black 
sheep I take away all the white ones, the black ones must 
remain, and if I have two kinds of propositions, those which 
are consistent with the premises and those which are incon- 
sistent with the premises, and I eliminate all those which are 
inconsistent with the premises, those which are consistent with 
the premises must remain. 

103. The process of making all the imaginable propositions 
which can be made out of the terms of given propositions and 
of then eliminating the inconsistent propositions, is the reason- 
ing process for non-numerical propositions. 

104. Suppose we have the proposition "Salt is chloride of 
sodium," the two terms are "salt" and "chloride of sodium," 
and if we posit these two terms, then by the Law of Opposites 
we must also posit "not-salt" and "not-chloride of sodium." 
These four terms which we now have can be combined into four 
propositions : 

1, "Salt is chloride of sodium;" 

2, "Salt is not-chloride of sodium;" 

3, "Not-salt is chloride of sodium ;" 

4, "Xot-salt is not-chloride of sodium." 

These four propositions can be read either way, forward or 
backward. Now, if "Salt is chloride of sodium," then the 
combination which says "Salt is not-chloride of sodium'' cannot 
be true by the Law of Contradiction, which says that a thing 
cannot be and not be at the same time, and if "Chloride of 
sodium is salt," — which is No. 1 read backward — then the pro- 
position which says that "Chloride of sodium is not-salt" is 
inconsistent, and by the Law of Contradiction must be elimi- 
nated. We have now left No. 4 which says that "Not-salt is 
not-chloride of sodium." This is not inconsistent with the 



32 THE LAWS OF THOUGHT. [Chap. 3. 

proposition that "Salt is chloride of sodium/' or with the pro- 
position that ''Chloride of sodium is salt."' As it is a consistent 
proposition we cannot eliminate it. Therefore from the propo- 
sition that "Salt is chloride of sodium/' we have found a new 
proposition, viz.: that what is not-salt is not-chloride of 
sodium. This is an equivalent proposition to the proposition 
tjiat "Salt is chloride of sodium.'' For if we take this conclu- 
sion for our premise, "Not-salt is not-chloride of sodium," we 
can get for our conclusion "Salt is chloride of sodium," which is 
our original premise. 

105. Thus, if we posit the terms "not-salt" and "not-chlo- 
ride of sodium," we must also posit the terms "salt" and "chlo- 
ride of sodium." These four terms give us four propositions, 
viz.: 

1, "Not-salt is not-chloride of sodium;" 

2, "Not-salt is chloride of sodium ;" 

3, "Salt is not-chloride of sodium;" 

4, "Salt is chloride of sodium." 

Now if No. 1 is true, "Not-salt is not-chloride of sodium," then 
No. 2 "Not salt is chloride of sodium" is inconsistent and must 
be eliminated. Now if No. 1 is true when read backward "Not- 
chloride of sodium is not-salt," then No. 3, which is "Not-chlo- 
ride of sodium is salt" when read backward, is inconsistent and 
must be eliminated. No. 4, "Salt is chloride of sodium, 1 ' 
remains, and as that is not inconsistent with No. 1, "Not-salt is 
not-chloride of sodium," it stands. And thus from the propo- 
sition "Not-salt is not chloride of sodium," we have returned 
to the proposition with which we started, "Salt is chloride of 
sodium." 

If anyone should fail to see that the proposition "Salt is 
chloride of sodium" can be read backward as well as forward, 
it can be easily demonstrated by using the Law of the Excluded 
Middle, thus: "Chloride of sodium is either salt or it is not- 
salt." If we suppose that it is not salt, then since by our pre- 
mise "Salt is chloride of sodium, " salt would be not-salt, which 
is impossible according to the Law of Contradiction, therefore 



§§ 106, 107.] LAW OF COMBINATIONS. 33 

chloride of sodium must be salt. And, again: if not-salt is 
not-chloride of sodium, then by the Law of the Excluded Middle 
not-chloride of sodium must be either not-salt or salt. But if 
we suppose not chloride of sodium to be salt, then since not-salt 
is not-chloride of sodium, not-salt would be salt, — which is 
absurd. Therefore not-chloride of sodium must be not-salt. 

106. The Law of Combinations in logic is a different law 
from the Law of Permutations in mathematics. The Law of 
Combinations in logic pays attention only to the fact that 
terms are positive or negative, and does not pay attention to 
the order in which they are read. In logic, as in algebra, it 
makes no difference whether we say x is y, or y is x. But the 
Law of Permutations is concerned with the order in which 
terms can be read. The Law of Combinations doubles the 
number of combinations with each additional letter. Its for- 
mula is 2 x 2 x 2 x 2, etc. Two letters make four combina- 
tions, three letters make eight combinations, four letters make 
sixteen combinations, five letters make thirty-two combina- 
tions. The formula of permutations is, 2 x 3 x 4 x 5, etc. Thus, 
two letters can be read in two ways, three letters in six ways, 
four letters in twenty-four ways, and five letters in a hundred 
and twenty ways. 

107. Suppose we have the two letters A and B, we can make 
four combinations as follows: AB, Ab, aB and ab. If we 
have three letters ABC, we can make the following combina- 
tions: ABC, ABc, AbC, Abe, aBO, aBc, abC and abc. The 
method pursued in making these combinations is simply 
changing one letter at a time. This will give us every possible 
combination. It is more convenient to commence to change 
the last letter first and proceed backward until every letter 
has been changed. These combinations are to be considered 
as propositions which can be read in any order we choose. 
Thus, suppose a boy had three christian names, Andrew, Ben- 
jamin and Charles. We could say Andrew is Benjamin, or 
Charles is Andrew, or Benjamin is Andrew Charles, — in short, 

3 



34 THE LAWS OF THOUGHT. [ Chap. 3. 

we could read the names in as many different ways as there are 
permutations and combinations, and each and every one of the 
propositions thus derived will be the equivalent of each and 
every one of the others. We can choose any one of these 
equivalent expressions which suits our convenience. 



CHAPTER IV. 

INFERENCE. 

108. Inference is of two kinds, immediate and mediate. 
Immediate Inference is where one proposition being given to 
the mind, the mind immediately perceives the truth of another 
proposition. Mediate Inference is where two or more propo- 
sitions are given to the mind and from them the mind infers 
other propositions. This is the view of Inference which is 
taken by the old logic. It is doubtful to me whether there are 
any immediate inferences except those w T hich w r e call Axioms 
and Laws of Thought. 

109. My view of Inference is that it is the result of a process 
by which we make all the possible new propositions which can 
be made out of the terms of the given propositions, and then 
eliminate those propositions w T hich are inconsistent with the 
premises. The consistent propositions remain. 

110. According to the old logic, from a proposition in the 
form of "All S is P," by Immediate Inference we can get "Some 
P is not-S." When we try the experiment of asking persons 
who are not skilled logicians whether they can. immediately 
infer "Some P is not-S" from "All S is P," the experiment 
proves that the great majority of men cannot see the Imme- 
diate Inference. I do not think myself that there is any such 
Immediate Inference. In the first place "All S is P," which 
means "All S is some P," is not a true proposition. "Some P" 
is not a definite and synonymous name for "All S." To take a 
concrete example: "All men are animals." It is easy to prove 
that animals is not a true name for men, because when we use 
the Law of the Excluded Middle and say "Animals are men or 
not-men" we come to a stop. We must first convert the propo- 
sition "All men are animals" into a definite proposition, "All 
men are animal-men," and then we can immediately infer that 



36 INFERENCE. [Chap. 4. 

"Animal-men are men." But if anyone should hesitate, it can 
be proved by the Law of the Excluded Middle, thus: "Animal- 
men are men or not-men." If we suppose animal-men to be 
not-men, then since men are animal-men, men would be not- 
men, — which is absurd according to the Law of Contradiction, 
therefore "Animal-men are men." 

111. Whenever we have a proposition whose subject and 
predicate are tautologous or synonymous, we can get a new 
proposition by reading it backward. This is a case of Imme- 
diate Inference. 

112. Some of the old logicians think that we can have Imme- 
diate Inference by qualifying both the subject and predicate by 
the same term, thus: "A negro is a fellow creature," therefore 
"A suffering negro is a suffering fellow creature." But when 
we construct parallel examples we see that this method is 
illogical. Thus: "An elephant is an animal." Therefore "A 
small elephant is a small animal;" "A cricketer is a man," 
therefore "A poor cricketer is a poor man." By converting the 
above indefinitepropositions intotruepropositions, such as "Ad 
elephant is an animal-elephant;" "A cricketer is a man- 
cricketer," then we can say, "A small elephant is a small ani- 
mal-elephant," "A poor cricketer is a poor man-cricketer," and 
our inferences will be true. 



CHAPTER V. 

SIGNS. 

113. In complicated reasoning we require the help of signs. 
Of course it would be possible to write out all the propositions 
which can possibly be made out of the terms of given propo- 
sitions, at full length. But it would be a very laborious and 
tedious task. Use of appropriate signs will save us a great 
deal of time and trouble. The success of the mathematical 
sciences is due largely to the fact that they have an appro- 
priate language of signs. We shall need signs to stand for 
terms, for the copula, for "or," and for the Universe of Dis- 
course. 

114. The sign for the copula which I shall use is two short 
parallel lines, thus: ==. I use this sign because it has been 
heretofore generally used to stand for "is," but at the same time 
I wish to say that it must not be confounded with the similar 
sign used in mathematics. In logic it means "has the name 
of;'- in mathematics it means "is equal to." These two mean- 
ings are distinct. 

115. For the word "or" I shall use a perpendicular line, 
thus : |. Of course this is arbitrary. 

116. The letters of the alphabet are probably the best signs 
we can use to stand for terms. We shall need two kinds of let- 
ters, one to represent positive terms and one to represent nega- 
tive terms. We shall use the capital letters, A, B, C, etc., to 
stand for the positive terms, and the small letters, a, b, c, etc., 
to stand for the negative terms. Thus, if A stands for man, a 
will stand for not-man; if B stands for straight, b will stand for 
not-straight; if C stands for Charles, c will stand for not- 
Charles. 

117. We do not need any sign for a conjunction. The jux- 
taposition of two letters will enable us to supply the conjuuc- 



33 SIGNS. [Chap. 5. 

tion which will best suit our convenience. Thus, AB can be 
read "A and B," when we wish to indicate that they are to be 
taken together. If we have several letters like ABCD we can 
read them "A, B, C and D." Our letters will stand for the 
names of ideas. When the names are either adjectives or com- 
mon nouns we must convert them into definite arid svnono- 
mous terms before representing them by signs. When con- 
verted we can represent them by either one or two signs, or 
even more, as best suits our convenience. Thus: if A stands 
for man, we can use BA to stand for animal-man. Take the 
proposition "The British Museum is the largest collection of 
books in the world." If A stands for the British Museum, we 
can let B stand for "the largest collection of books in the 
world." 

118. Usually it will only be necessary to use a single letter 
to stand for a subject or a predicate, except in those cases 
where we have indefinite subjects or predicates or compound 
subjects or compound predicates, or both. Where we have a 
compound subject or a compound predicate, each elementary 
term which goes to make up the compound term can be, and 
generally should be, represented by a separate letter. With 
these four kinds of signs, viz. : capital letters, small letters, the 
copula and the sign for "or," we can represent all necessary 
logical distinctions. 

119. The copula, though necessary from a grammatical 
point of view, is not necessary from a logical standpoint. 
When the baby says "papa, man," it utters a true proposition, 
and in our system of logic after having once formally stated a 
proposition we shall dispense with the use of the sign =f or the 
copula, and simply write the letters one after the other. Thus 
"AB" will mean "A is B, and B is A." When we have more 
than two letters we can insert the copula wherever we please. 
Thus, when we have "ABC," by inserting the copula wherever 
we please, we can get a great variety of propositions, but, of 
course, they will only be equivalent propositions, because 
"ABC" means simply that a certain object has three names, 
viz. : A, B and C. 



§§ 120-125.] DISJUNCTIVES. 30 

120. In our system a letter stands for a name, whether that 
name be a designation, description, or proper name. The name 
stands for an idea, a thought; the idea must be definite, either 
an individual or a class or a collection, or two or more tilings 
taken together as one thing, or an indefinite number of things 
taken together as one thing. In other words, the name must 
represent either a thing or a bundle, no matter how many 
things are contained in the bundle. 

121. In our system we pay no attention to many of the old 
distinctions of the syllogistic system, such as denotation and 
connotation, essential and accidental, exclusion and inclusion, 
etc. 

122. The reason why we have no signs to indicate a denial 
is because denial is no part of logic. 

123. We are not obliged to read our letters in the order 
ABCD, etc. Of course we are obliged to put one letter before 
another in speech, but in logic they all stand on an equality 
and it is just as logical to read them DCBA as it is to read 
them ABCD. 

121. The proposition "A is B or C" is written "A = Be | 
bO." By the law of Opposites, whenever we posit a thought we 
are also obliged to posit its negative, and as the Avoid "or'' is 
exclusive the proposition "A is B or C" really means A is B and 
not-C or C and not-B, that is, it is one or the other, but not both, 
and in order to express this meaning clearly and definitely by 
our signs we are obliged to express it A = Be | Cb. To put it 
in another light: in stating alternatives by means of signs we 
must give the full and complete symbolic description of each 
alternative. Thus: A is B or C or D is stated thus: A = 
ABcd | ACbd | ADbc; this gives us a perfect symbolical des- 
cription of each alternative. In stating alternatives it is better 
practice to repeat the subject in the predicate unless the con- 
trary is indicated. This is necessary when we do not know 
whether in the proposition A is B or or D, that either B or C 
or D is synonymous with A. 

125. Whenever we make a proposition we always have in 



40 



SIGNS. 



[Chap. 5. 



view what is technically called the Universe of Discourse, that 
is, our field is a more or less limited one, and our remarks are 
made with reference to the subject which we have under con- 
sideration; thus our field of discourse may be business or law 
or logic or everything which is conceivable by the mind, out 
whatever our Universe of Discourse may be, our propositions 
are to be taken with reference to it. Now we need a sign to 
represent this Universe of Discourse. The sign which I have 
chosen is a square. I prefer a square because it can be divided 
and subdivided into smaller sections indefinitely, at least the 
only limits are physical limits. Now when I want to repre- 
sent A and a I make a square, thus: 



Fig. 1. 



This square represents the Universe of Discourse. Now 
draw a perpendicular line through the center of it which will 
divide it into two parts, and over one part write "A" and over 
the other part write "a," thus: 



Fig. 2. 



Now we have represented A and a visually by means of a dia- 
gram. Now if we want to divide our Universe of Discourse 
still further into B and b we will draw a line horizontally 



§ 120.] 



UNIVERSE OF DISCOURSE. 



41 



through the center of our square and mark the upper part B and 
the lower part b, thus : 



Fig. 3. 

12G. The sections which run from right to left we will, for 
the sake of convenience, call rows, and the sections which run 
up and down we will call files. Thus we have A files and a 
files and B rows and b rows. Each of our sections thus 
obtained will represent a conceivable idea, and its name will be 
the letters which would meet in that section. Thus, the sec- 
tion where A and B would meet will be the AB section and it 
means that an imaginable idea is described by the names A and 
B. or to put it in the form of a proposition, "A is B, or B is A." 
The file section below AB is the Ab section ; it means A is b, or 
b is A. The row section to the right of AB is the aB section 
and it means a is B, or B is a, whichever way we choose to put it. 
The remaining section is the ab section, and it means a is b, or b 
is a. If we wish to do so we can repeat a letter and read our 
propositions thus: A is BA, B is AB; a is ba, etc. It will be 
understood that the file letters A and a which stand at the head 
of the file columns run all the way down, that is they are to be 
repeated in each section under them, and the row letters B 
and b which stand at the right of the row sections run. all the 
way across and are to be repeated in each of those sections. 
Our diagram will now have this appearance: 



42 



SIGNS. 



[ Chap. 5. 



AB 


aB 


Ab 


ab 



Fig. 4. 



127. Now this Diagram represents that our Universe of Dis- 
course is divided into four sections and each section represents 
a conceivable idea. Given the terms A and B we must posit 
their opposites a and b, and our diagram now represents every 
conceivable combination which can be made with our terms. 
It is a practical illustration of the process of dichotomy. The 
two terms A and B produce the four combinations and eight 
propositions represented in our diagram. 

12S. I call this diagram a Reasoning Frame, because I 
believe that with its aid the process of reasoning can be repre- 
sented visually, the combinations can be made mechanically 
and the elimination of inconsistent propositions from the 
Frame can also be done in a mechanical manner, and then the 
propositions which are consistent with the premises will auto- 
matically remain. 

129. I believe that the brain is a thinking machine, and this 
system represents the mechanical nature of the brain's activity 
in the reasoning process. 

130. Suppose our proposition is "A is B." Xow we know by 
the Law of the Excluded Middle that B is A or a. If we sup- 
pose B to be a, then since A is B, A would be a, which is impos- 
sible by the Law of Contradiction. Therefore B is A. 

Xow if A is B, then the combination Ab, which means that A 
is b, is inconsistent with the proposition A is B, and we elimi- 
nate it from our Reasoning Frame by making a figure 1 in that 
section. The sign 1 indicates that we have eliminated an incon- 
sistent proposition. Xow if B is A then the combination Ba, 
which means that B is a, is an inconsistent combination and we 



§§ 131, 132.] 



UNIVERSE OF DISCOURSE. 



43 



eliminate it by making a figure 2 in the aB section. The sec- 
tion ab, which means a is b, is not inconsistent with the pro- 
position A is B, and we cannot eliminate it. It therefore auto 
matically remains. 

131. By our process we have now ascertained that the pro- 
position that A is B is consistent with three other propositions, 
viz.: B is A, a is b, and b is a. We may start with either one 
of them as a premise and we will get the other three as conclu- 
sions. Our diagram, after performing the above operation, will 
have this appearance: 



AB 


aB 


1 
Ab 


ab 



Fig. 5. 

132. Suppose to make this operation clearer we take a con- 
crete example. Take the proposition, "Salt is chloride of 
sodium." This is a proposition in which the terms are synony- 
mous and we can read it backward, viz. : Chloride of sodium 
is salt. 

Let A stand for salt 

B stand for chloride of sodium, then 
a will stand for not-salt 
b will stand for not-chloride of sodium. 
The proposition can be stated, thus: 

A = B. 
Xext we make our square, thus: 



Fig. G. 



44 



SIGNS. 



[ Chap. 5. 



Then we divide it into A and a, thus: 

A a 



Fig. 7. 
Next we divide it into B and b, thus; 

A a 



Fig. 8. 
Then we mark our sections, thus: 

A a 



AB 


aB 


Ab 


ab 



Fig. 9. 

Now if A is B, Ab is an inconsistent proposition and we elimi- 
nate it by making a figure 1. Again, if B is A, Ba (aB) is an 
inconsistent proposition and we eliminate it by making a figure 
2. But the combination ab is not inconsistent with our pre- 
mise, which is A is B, and we cannot eliminate it. It there- 
fore remains, and all that remains for us to do is to translate 
it into concrete terms. 



133.] 



UNIVERSE OF DISCOURSE. 



45 



Thus, "What is not-salt is not-chloride of sodium; what is 
not-chloride of sodium is not-salt. Our diagram has this 
appearance after having performed these operations: 

A a 



AB 


•> 
aB 


1 
Ab 


ab 



Fig. 10. 



The propositions which we struck out on account of their 
inconsistency with the premise could be translated, Salt 
is not-chloride of sodium, and Chloride of sodium is not-salt. 
If our premise is true then by the Law of Contradiction these 
two propositions, Salt is not-chloride of sodium, and Chloride of 
sodium is not-salt, cannot be true. 

133. Next let us take the proposition "Chloride of sodium 
is salt." 

Let B = chloride of sodium, 
A = salt ♦ 

b = not-chloride of sodium 
a = not-salt 
Then the premise will be stated thus: 

B = A. 
Then draw a square, divide it into two parts by a perpen- 
dicular line and over the left hand part write A and over the 
right hand part w T rite a; then divide it again by a horizontal 
line into equal parts, and at the right hand of the upper part 
write B, and at the right hand of the lower part write b. Our 
square is now divided into four sections, each section repre- 
sents an imaginable idea. These are all the imaginable ideas 
which can be obtained from two terms, A and B. Next name 
each section with the file letter over it and with the row let in- 
to the right of it. Our sections will then be named AB, Ab, 



4G 



SIGNS. 



[ Chap. 5. 



aB, ab. These combinations mean that each imaginable idea 
lias two names, and these two names w r ill make tw T o imaginable 
propositions. Thus AB means A is B and B is A. Our square 
will have this appearance after the operation. 

A a 



AB 


1 
aB 


2 

Ab 


ab 



Fig. 11. 

In removing the inconsistent propositions from our square or 
Reasoning Frame it is not necessary for us to pay attention to 
our concrete proposition, "Chloride of sodium is salt;-' it is only 
necessary to pay attention to our abstract signs which repre- 
sent that proposition, viz. : "B is A." 

Now if B is A it follows by the Law of the Excluded Middle 
that A is B, because A is B or b. Now if we suppose A to be b, 
then since B is A by the premise, B would be b, which is impos- 
sible by the Law of Contradiction, therefore A is B. 

Now if B is A, then the combination Ba, which means that 
Chloride of sodium is not-salt, is inconsistent and we eliminate 
it by placing a figure 1 in the Ba section. Again, if A is B, the 
combination Ab, which means that Salt is not chloride of 
sodium, is inconsistent and we eliminate it by a figure 2 in the 
Ab section. The section marked ab remains, and it is not 
inconsistent with AB. Its translation into concrete terms is, 
What is not-salt is not chloride of sodium, and What is not- 
chloride of sodium is not-salt, and of course both of these propo- 
sitions are true if the premise "Chloride of sodium is salt" is 
true, since we have used nothing but infallible laws of thought 
to deduce them. The system is infallible no matter whether 
the premises are so or not. 

1 81. According to the old logic, no inference could be drawn 
from two negatives. This is another of its mistakes. Let us 



§ 135.] 



UNIVERSE OF DISCOURSE. 



4 7 



take the proposition "What is not-salt is not- chloride of 
sodium." 

Let a = not-salt 

b = not-chloride of sodium 

A = salt. 

B = chloride of sodium 
Make a square, divide into two equal parts by a perpendicu- 
lar line, then into two equal parts by a horizontal line; over the 
files w T rite respectively A and a, to the right of the rows write 
respectively B and b; mark each section with the letters which 
would meet in that section. Our square will have this appear- 
ance after the operation: 

A a 



AB 


1 
aB 


2 
Ab 


ab 



Fig. 12. 

Our concrete proposition, "What is not-salt is not-chloride 
of sodium," can be stated thus, a = b. 

Now if a is b it follows by the Law r of the Excluded Middle 
that b is a, because b is either A or a. But if we suppose b to be 
A, then, since a is b, a would be A, which is impossible by the 
Law of Contradiction, — therefore b is a. 

Now if a is b, the proposition a is B, aB, is inconsistent and 
we eliminate it with a figure 1. If b is a then the combination 
bA is inconsistent and we eliminate it with a figure 2. The 
combination AB remains. It is not inconsistent with the 
premise, ab, and it can be translated "Salt is chloride of 
sodium." 

135. Thus, from the proposition a is b we have deduced the 
affirmative proposition A is B and this shows that in a correct 
logic you can reason as accurately and as easily with negative 
terms as you can reason with positive terms. 



48 SIGNS. [ Chap. 5. 

9 

136. Next let us take the proposition, "What is not-chloride 
of sodium is not-salt." 

Let b = not-chloride of sodium 
a = not-salt 
B = chloride of sodium 
A = salt / 

Our premise can be stated thus: 

b = a. 
Make an AB diagram. 

A a 



AB 


2 
aB 


1 
Ab 


ab 



Pig. 13. 



If b = a, then a will equal b, because by the law of the 
Excluded Middle a = B | (or) b. But if we suppose a to equal 
B, then, since b is a, b would be B, which is impossible by the 
Law of Contradiction. Therefore a is b. 

Now if b is a, then the combination bA is inconsistent and we 
mark it with a figure 1. Again, if a is b, then the combination 
aB is inconsistent and we mark it with a figure 2. The com- 
bination AB is not inconsistent with the premise, and it auto- 
matically remains in the Reasoning Frame, and it can be trans- 
lated, "Salt is chloride of sodium or Chloride of sodium is salt.'' 

137. We have now shown that given any simple categorical 
proposition of the AB type with synonymous terms there are 
four ways of stating it, viz.: AB, BA, ab, and ba. This fur- 
nishes us with an easy method of testing the truth of simple 
propositions, for if any one of the four equivalent ways in which 
the proposition can be stated, is untrue, then the premise is 
untrue. A simple proposition when first stated may appear to 
be true, but if we convert it into its equivalent forms, if it is 



§§ 138-142.] 



THE REASONING FRAME. 



49 



not true, one of the equivalent forms will usually show us that 
the proposition is false. 

138. This diagram for two terms (and the examples which 
we have given of its proper use), is the key to our system. It 
can be extended to any number of terms. All that we will 
have to do is to divide our Universe of Discourse into as many 
sections as there are imaginable combinations of the terms of 
our premises, and then eliminate the inconsistent combinations. 

139. The process of combining the terms so as to make all 
the imaginable combinations, can be done in a purely mechani- 
cal manner. The making of the Seasoning Frame is also 
mechanical. The eliminating of the inconsistent combinations 
is a mechanical process. When we have done this, our con- 
clusions, and the other consistent combinations, automatically 
appear. 

140. Xo matter how many letters there may be in any sec- 
tion, they always indicate that they are the names of one idea. 

141. If we start with two terms we shall have, by the Law 
of Combinations, four combinations, and every additional letter 
will double the number of combinations, three terms make 
eight combinations; four terms make sixteen, and so on. In 
our Reasoning Frame we must have a section for each combi- 
nation." It follows necessarily that each section will have a dif- 
ferent combination of names from every other section. 

142. Let us make a square and bisect it with a vertical line, 
and mark one section A and the other a, thus: 



Fig. 14, 



50 



SIGNS. 



[ Chap. 5. 



We have now an illustration of the Law of Relativity. We 
perceive that A differs from a, and that a differs from A. The 
diagram also illustrates the Law of Opposites, that if we have 
A we must also have a, and if we have a we must also have A. 
We can make a hypothetical proposition out of it, thus : If A, 
then a, and if a, then A. 

Id8. The Law of Identity is also exemplified. It says A is 
A, and a is a, and it means if a thought has the name of A, then 
it has the name of A, and if an idea has the name of a then it 
has the name of a. 

144:. The Law of Contradiction can also be found repre- 
sented in the diagram, which says that the same thing cannot 
be A and a at the same time and place. It means that the same 
thing cannot have two opposite names. 

145. The Law of the Excluded Middle is also illustrated. It 
says that anything is either A, or it is a, and it means that 
auything either has a given name, or it has the opposite name. 

140. Now let us make a square and bisect it with a vertical 
line and then bisect it with a horizontal line, and over the files 
write A and a, and against the rows mark B and b, respectively, 
and mark each section with the letters that would meet in it, 
thus: 

A a 



AB 


aB 


Ab 


ab 



Fig. 15. 



147. By the Law of Relativity we perceive each section is 
different from every other section. 

148. By the Law of Opposites, for every A there is a and for 
every B there is b, and conversely. 

149. By the Law of Identity we can read AB is AB, aB is 
aB, and so on. 



g| 150-153.] 



THE REASONING FRAME. 



51 



150. By the Law of Contradiction we perceive that A can- 
not be AB and Ab at the same time and place, and so on. 

151. We can also read the Law of the Excluded Middle, for 
A is either AB or Ab, and a is either aB or ab, and so on. 

152. Thus we have illustrated in the Reasoning Frame for 
two terms all the laws which are necessary to use in the most 
complicated logical reasoning. Of course a Reasoning Frame 
is not logic, but it is rather a foundation for logic and it enables 
us by its visual demonstration of logical processes to more eas- 
ily comprehend a rather abstruse science, but still, one that is 
not so abstruse as arithmetic or algebra. 

153. Suppose we have the proposition "Death is not life," 
and we wish to know what inferences can be drawn from this 
premise. 

Let A = Death, 

b = not-life, 

a = not-death, 

B = life. 
Make an AB diagram, as heretofore described, thus: 



1 

AB 


aB 


Ab 


2 
ab 



Fig. 16. 



Our premise can be stated, thus: 

A is b 

Then if A is b the combination AB, which says that A is B, is 
inconsistent and we eliminate it by making a figure 1 in that 
section. 

Again, if A is b, b is A by the Law of the Excluded Middle, 
for b is either A or a. But if we suppose b to be a, then, since 



52 



SIGNS. 



[ Chap. 5. 



A is b bv the premise, A would be a, which is impossible by 
the Law of Contradiction, therefore b is A. 

Now, if b is A, then the combination ba, which means that b 
is a, is inconsistent, and we eliminate it by making a figure 2 
in that section. The combination aB is not inconsistent with 
the premise Ab; it automatically remains in the Reasoning 
Frame and all we have to do is to translate it into its proper 
concrete terms, viz.: "Not-death is life," or "Life is not- 
death." 

154. Supposing now that we take the proposition "Life is 
not-death." In stating propositions we take capital letters to 
stand for positive terms, and we take negative letters, i. e., 
small letters, to stand for negative terms. The object in doing 
this is to prevent us from becoming confused in the use of our 
symbols. We could if we chose, let a capital letter stand for 
a negative term, but the probability is that in so doing we 
should fail to remember that our positive letter was standing 
for a negative term, and thus make a mistake in manipulating 
our symbols. To return to our premise. 

Let B = life, 

a = not-death. 
b = not-life. 
A = death. 

The premise can be stated thus: 

B is a 

Make an AB diagram as heretofore described, thus: 



1 

AB 


aB 


Ab 


2 
ab 



Fig. 17. 
Now if B is a, then the combination BA, which means 'that 



§ 155.] THE REASONING FRAME. 53 

B is A, is inconsistent, and we eliminate it by making a figure 
1 in that section. 

If B is a, then a is B by the Law of the Excluded Middle, 
because a is either B or b, and if we suppose a to be b, then, 
since B is a, by the premise, B would be b, which is impossible 
by the Law of Contradiction, therefore a is B. 

If a is B, then the combination ab, which means a is b, Is 
inconsistent, and we eliminate it b} T making a figure 2 in thai 
section. The combination bA is not inconsistent with the 
premise, Ba, and it therefore automatically remains in the 
Reasoning Frame and it can be translated "Not-life is death." 

155. There is a shorter method of working these examples 
than the one which we haye been pursuing. If the reader will 
look back over the examples which we have worked he will 
see that when our premise was the combination AB, then the 
combination in the same file, viz.: Ab was inconsistent, and 
the combination in the same row, viz.: aB was inconsistent. 
Again, when our premise was Ab the combination in the same 
file, viz.: AB was inconsistent, and the combination in the 
same row, viz.: ab was inconsistent. Again, when our pre- 
mise was aB, then the combination in the same file, viz.: ab 
was inconsistent, and the combination in the same row, viz.: 
AB was inconsistent. Again, when our premise was ab, then 
the combination in the same file, viz.: aB, was inconsistent, 
and the combination in the same row, viz.: bA, was inconsis 
tent. From these examples we can see that the combinations 
in the same row and in the same file with our premise, are 
always inconsistent in the case of propositions of the AB type, 
where the terms are synonymous; and the inconsistent pro j po- 
sitions can be eliminated by this rule without stopping to rea- 
son out the matter. 

Suppose our premise is "Not-death is life.' > 
Let a = not-death. 

B = life, 

A = death, 

b = not-life. 



54 



SIGNS. 



[ Chap. 5. 



Our premise can be stated, thus: 

a = B 
Make an AB diagram as hereinbefore directed, thus; 



2 
AB 


aB 


Ab 


1 
ab 



Fig. 18. 

Now as aB is our premise, by the rule laid down in the last 
preceding section, we can eliminate the ab section (1), because 
it is in the same file, and the AB section because it is in the 
same row (2); the Ab combination remains and its translation 
is ''Death is not-life." 

150. If our premise is "Not-life is death," 
Let b = not-life, 
A — death, 
B = life, 
a = not-death. 
Our premise can be stated, thus: 

b = A 
Make an AB diagram, thus: 



1 

AB 


aB 


Ab 


2 

ab 



Fig. 19. 

Now as our premise is the bA combination, then the combi- 
nation in the same file, viz. : AB, is inconsistent, and we elimi- 



§§ 157-159.] REASONING. 55 

nate it, and the combination ab in the same row is inconsistent 
and we eliminate it, as shown in the diagram above. We thus 
see that there are four equivalent ways of stating a propo- 
sition having synonymous terms which can be reduced to the 
Ab form, viz.: Ab, Ba, bA and aB. By taking any one of 
these for a premise we get the others. If any one of the equiva- 
lent propositions should as a matter of fact be false, that 
would prove that our premise was not true. 

157. By our system reasoning is reduced to the repetition 
of a few uniform operations of analyzing our proposition into 
terms, of representing the terms by appropriate signs, of mak- 
ing all the possible logical combinations and of eliminating the 
contradictory propositions. The most complicated questions 
can be solved in this routine manner when we understand the 
meaning of the premises. Of course if we do not understand 
the meaning of the terms used in the premises we cannot rea- 
son about what we do not comprehend. We must know what 
the terms mean before we can state them by means of signs. 

158. Most all logicians of the old school have had great dif- 
ficulty in interpreting negative terms and of deducing the con- 
sequences which follow from them. De Morgan asks how 
many persons would be able to say confidently and off-hand 
whether either, and if so which, of these two statements is 
true? (1) The English who do not take snuff are included in 
the Europeans who do not take tobacco. (2) The English who 
do not take tobacco are included in the Europeans who do not 
take snuff. (Snuff-takers of course are included in tobacco 
takers.) Or, again, Who are the non-ancestors of all the non- 
descendants of A. B? 

159. Prof. Venn in his excellent work on Symbolic Logic, p. 
21, speaking of logical diagrams, says: 

"A way of interpreting and arranging propositions which 
may be substituted for both the preceding (for the purpose of 
an extended symbolic logic), is perhaps best described as 
implying the occupation or non-occupation of compartments." 
His idea seems to be that when a section is eliminated it means 



56 SIGNS. [ Chap. 5. 

that it is unoccupied. When we eliminate a combination we 
do so because there is no idea which has the names which are 
symbolized by the combination, and when we let a combination 
stand because it is consistent with the premise we do so on the 
theory that the premise being true there is a thought which 
has the names which are symbolized by that combination. 

100. Boole discovered an algebraical % system of logical 
inference which worked in a mechanical manner. His system 
is a great improvement on the old logic, but it is a very difficult 
system to master, and it takes much longer to reach the con- 
clusions which we can reach by our system in a few minutes. 

101. The Eulerian system of representing propositions by 
means of circles is well known to readers of the old logic. It 
is a very imperfect scheme, however, and is of little practical 
use. 

102. Kant and De Morgan both suggested the use of a 
square and a circle, one to represent the subject and the other 
to represent the predicate. R. G. Latham and a Mr. Leech- 
man, use a square, circle and triangle all in one figure. 

163. Bolzano used parallelograms. All of these diagrams 
were used to represent the inclusion and exclusion of classes 
according to the old logic. 

164. The best diagrams that I have read of are those made 
by Dr. Marquand of Johns Hopkins University. He makes the 
squares in the same way that I do. His system of lettering 
the sections is different. 

165. When I discovered this method in March, 1895, I was 
not aware of the progress that had been made by other logi- 
cians in solving logical problems with the aid of diagrams. 
Since then I have looked the matter up as far as I could and 
have been surprised at the extent to which diagrams have 
heretofore been used. Some logicians object strenuously to 
the use of diagrams. But those who believe in a mechanical 
system to represent the thinking process, will have no sym- 
pathy with the views of these writers. 

16G. rioucquet also made use of squares which he claimed 



§§ 167, 168.] 



INCONSISTENCY. 



to have invented prior to Enler's system, and Kinase was one 
of the first logicians to use diagrams. 

167. I have gathered these facts from Prof. Venn's work on 
Symbolic Logic. Prof. Venn uses ellipses which intersecl 
each other, for the purpose of illustrating the reasoning pro- 
cess. His system is a good one, but in comprehensiveness and 
facility of manipulation it is not equal to the system of 
squares. 

168. Our system is able to detect any inconsistency in the 
premises. When there is a contradiction in the premises ii 
will manifest itself by the disappearance from the Reasoning 
Frame of onV or more of the letters which we use as signs of 
the terms which represent the premises. Suppose we had for 
one premise ''Salt is chloride of sodium" and for another pre- 
mise "Salt is not-chloride of sodium." 

Let A = salt, 

B = chloride of sodium, 

a = not-salt, 

b = not-chloride of sodium. 
Our premises can be stated, thus: 

(1) A = B 

(2) A = b 
Now make an AB diagram, thus: 



3 


2 


1 


4 



Fig. 20. 



Now since A is B, the combination Ab in the same file is 
inconsistent and w T e eliminate it by making a figure 1. And 
the combination a is B in the same row is inconsistent and we 
eliminate it bv making a figure 2. And. secondly, since A is 



58 SIGNS. [ Chap. 5. 

b, the combination AB in the same file is inconsistent and we 
eliminate it by making a figure 3, and the combination ab in 
the same row is inconsistent and we eliminate it by making a 
figure 4. Thus we see that on account of using contradictory 
premises every letter term has disappeared from the Seasoning 
Frame. 

169. The fact that we eliminated the combination Ab 
because it was inconsistent with the combination AB, does not 
prevent us from using it as a premise for the purpose of ascer- 
taining whether there are any combinations in the Reasoning 
Frame which are inconsistent with it. 

170. Whenever in the working of propositions one or more 
of the letters used disappears from the frame we must stop 
right there. The Frame tells us that our premises are incon- 
sistent and that no inferences can be deduced from them. 

171. So far we have been considering simple propositions 
of the kind which are often used in definitions. They are 
very important and occur with great frequency in everyday 
life, and yet the old logic failed to recognize their importance 
or to provide a place for them. Aristotle w T ent so far as to say 
that singulars cannot be predicated of other terms. This 
means that an idea could not have two singular names. Such, 
for instance, as "salt" and "chloride of sodium." Of course 
any system founded on such mistaken, views of logic must be 
full of errors. 

EXAMPLES FOR PRACTICE. 

172. What inferences can be drawn from the following pre- 
mises? (1) The British Museum is the largest collection of 
books in one building in the world. (2) The capital of the 
Fnited States is Washington, D. C. (3) Matter is not spirit. 
(4) What is not fit to teach is not proper to learn. 



CHAPTER VI. 

INDEFINITE PROPOSITIONS. 

173. Indefinite Propositions are those in which the predi- 
cate term is not limited to the subject term. They do not 
clearly and definitely point to the subject. When I say, 
"George is wise," the term "wise" applies to a great many indi- 
viduals besides George; it is not limited to him, nor does it 
point him out as the person thought of. 

If "George is wise" is a logical proposition, then "wise" is 
George, by the Law of the Excluded Middle, for "wise" is 
either "George" or "not-George," but if we suppose "wise" to 
be "not-George," then since "George is wise" by the premise, 
George would be not-George. But this is impossible by the 
Law of Contradiction, therefore "wise is George" is true, if 
our premise "George is wise" is a true proposition. 

But we know as a matter of fact that "wise is George" is not 
a true proposition and therefore we must reject all proposi- 
tions in the form of "George is wise" unless we can reform 
them and make them true propositions. This we can easily 
do by adding the subject to the predicate, thus: "George is 
wise George." The correctness of this form is immediately 
perceived by the mind. 

174. A parallel case is where the predicate is a common 
term. Thus in the proposition "Iron is a metal," "metal" is a 
common term, and applies to a good many objects besides iron. 
The proposition "Iron is a metal" is not a logical proposition, 
because if it were logical we could read it backward and say 
"Metal is iron." We know by the Law of the Excluded Middle 
that metal is either iron or not iron; if metal is not iron, (lien 
since iron is metal, iron would be not iron, which is impossible. 
Therefore metal is iron. But we know as a matter of fact that 
the proposition ''Metal is iron" is not true, and therefore the 
proposition "Iron is metal" is not true in that form. But if we 



GO 



INDEFINITE PROPOSITIONS. 



[ Chap. 6. 



add the subject to the predicate and say that "Iron is metal- 
iron" we have a true proposition and one that can be read 
either way. 

J 75. The reason why propositions must be made so that 
they can be read either way is because the subject and predi- 
cate are both names for the one thought; they must be equally 
definite, and being names for the one idea they must be capable 
of transposition. 

176. The old logic calls this kind of propositions universal 
affirmative propositions. It says the meaning is that one class 
is included in a higher class. This is without any foundation 
in fact. If there is any inclusion about it, the implied mean- 
ing is that the quality indicated by the predicate is included 
in the subject, which is exactly the reyerse of the idea of the 
old logic. 

177. Boole in his system used the letter V to stand for the 
word "some'' Thus, "Iron is metal" would be stated by him 
A=VB, meaning, Iron is some metal. Liebnitz, Lambert, and 
Jeyons would state it in the form of 

A = AB 
meaning 'iron is iron-metal." This form is correct. Let us 
take the proposition "Iron is metal" and solye it by means of 
our diagramatic system. The proposition means "iron id 
metallic iron." 
Let A = iron, 
B = metal, 
then our proposition can be stated, 

A = BA 
Make an AB Reasoning Frame in the usual manner, thus: 



i 



Pis. 21. 



§§ 178-180.] DISJUNCTIVE DEFINITIONS. 61 

The premise A is BA is represented in the AB combination. 
As we know by the Law of Identity that A is A and B is B, we 
can repeat these letters as often as necessary without alter- 
ing the logical effect of our proposition. Now if A is BA, then 
by the Law of Contradiction the combination Ab, which means 
Unit A is b, is inconsistent and we make a figure 1 in that sec 
tion. There is no other combination in the Frame which is 
inconsistent with the premise. 

If we read our premise backward, BA is A, there is no other 
combination in the Frame which is inconsistent with it. 

We thus have the following combinations left in the Rea- 
soning Frame as consistent: (1) AB, (2) aB, (3) ab. We can 
read them as follows: A is B. which means "Iron is metallic- 
iron;" b is a, which means "what is not-metal is not-iron.*' 
Xow T in the case of B Ave have two combinations in which B 
appears, viz: BA and Ba, so that we cannot say that B is A 
alone, or that B is a alone. 

178. When a letter occurs in two combinations, like BA 
and Ba, we must read it thus: B is either BA | Ba, (the sign 
| means "or"), and we can translate it thus: "Metal is either 
metallic-iron or metallic not-iron," or in popular language 
"Metal is either iron or not-iron." Of course we know by the 
Law of the Excluded Middle that metal is either iron or not- 
iron, so that this information given to us in the Reasoning 
Frame about B, is of no especial value. It is, however, a true 
definition of B and it saves us from drawing any wrong infer- 
ences in regard to B. 

The letter a also appears in two combinations, aB and ah. 
Of course we cannot say that a is B alone, or that a is b alone, 
but we must read it a is either aB, | ab, which means that what 
is not-iron is either metal or not-metal. 

170. The only conclusions which we can regard of special 
value are those which are not in the disjunctive form, such as, 
b is a, in the present case. 

ISO. The old logic says that from the universal affirmative 



62 INDEFINITE PROPOSITIONS. [ Chap. 6. 

proposition all A is B, we can infer the particular proposition 
''Some A is B." This is extra logical. 

The old logic in order to sustain its position that some A is 
an inference from all A, says that "some" means "some and it 
may be all." This is an arbitrary meaning. 

181. The universal negative proposition in the old logic is 
symbol ized thus: "Xo A is B." It means that the object A 
has the name not-B. 

182. In reading our results where the subject is repeated 
in the predicate it is optional with us whether to translate it 
into concrete words or not. 

183. The particular affirmative proposition of the old logic 
has the word "some" in the subject, and the predicate term 
is generally undistributed, that is the whole of it is not 
taken, it is therefore indefinite. "Some men arewise'Ms a sample 
of the particular indefinite class. Now when we try to realize this 
proposition in our minds and ask what does the word "Some" 
in the subject mean, the answer must be that it means the wise 
men who are referred to in the predicate. "Some men" there- 
fore means in this case some men who are wise or some wise 
men. And if we ask what the adjective "wise" in the predi- 
cate qualifies, the answer must be that it qualifies the "Some 
men" referred to in the subject. Therefore the proposition 
"Some men 'a re wise" really means "wise men are wise men." 

If we let A = men 
B = wise 
then the proposition could be stated, 

AB = AB 

184. In such absurd propositions as "Some horses are 
dogs," the proposition "Some horses are dogs" means really 
"dog-horses are dog-horses." The proposition is true although 
we cannot realize the idea of some dog-horses. Xo one can 
deny the proposition that "dog-horses are dog-horses" even if 
he has not the slightest idea what the term "dog-horse" means. 
As we have said before, logic has nothing to do with the truth 
of our premises, its sole function is, given the premises to 



§g 183-186.] PARTICULAR PROPOSITIONS. 63 

deduce all the latent and implied meanings contained in them. 

185. It seems strange that the old logic should have given 
such a prominent place to particular affirmative propositions 
in which the subject is qualified by the word "some," and 
should have rejected the equally valid propositions in which 
the subject is qualified by the adjectives "few," "many," or 
"most." There is no reason in this arbitrary exclusion of 
"few," many" and "most." To my mind, "few," "many" and 
"most," are just as good words as "some." If "few, many and 
most" are to be rejected, certainly "some" should accompany 
them. 

186. Particular negative propositions are admitted by the 
old logic as valid logical propositions and are given a promi- 
nent place in that system. "Some elements are not metals" 
may be taken as an example of this type of proposition. When 
we come to ask ourselves what do we mean by the term "Some 
elements" described in the premise, the answer must be that 
we mean by "Some elements" the not-metals referred to in the 
predicate. The subject of the proposition therefore really 
means "Elements which are not metals," or to put it in other 
words, "elementary not-metals." Xow it is clear that the term 
"not-metals" used in the predicate refers to the "Some ele- 
ments" used in the subject. To make our proposition definite, 
therefore we must put it in this form: "Elementary not-metals 
are elementary not-metals. If we let 

A = elements 
b = not-metals 
the proposition can be stated in the form of 

Ab = Ab 



CHAPTER VII. 

SIMPLE CATEGORICAL TROrOSITIONS INVOLVING THREE TERMS. 

1ST. We have seen that the two terms A and B will yield 
four combinations, viz.: AB, Ab, aB and ab. Now when we 
add a third term C, we can divide each of these combinations 
into C and c. This will give us the following combinations: 

(1) ABC 

(2) ABc 

(3) AbC 

(4) Abe 

(5) aBC 
Mil aBc 

(7) abC 

(8) abc 

Thus three terms give us eight combinations, which we have 
obtained by dividing each of the four combinations of AB, Ab, 
aB. and ab, into two divisions, the division which is C, and 
the division which is c. 

188. There are two ways in which we can make a diagram 
containing eight sections. A square which is divided into two 
sections by four sections will of course give us eight sections. 

Our square may have its two sections run perpendicularly 
and its four sections run horizontally, or it may have its 
four sections run perpendicularly, and its two sections run 
horizontally. There is no logical difference between the two 
plans, but I have found the latter plan a little more conven- 
ient in practice. Let us make one of that kind first. Make 
a square, thus: 



§ 180.] 



THE REASONING FRAME. 



65 



Fig. 22. 

Then divide it into four perpendicular sections by three equi- 
distant vertical lines, thus: 



Fig. 23 
Then bisect these four sections by a horizontal line, thus 



Fig. 24. 

We have now eight sections, representing our eight com- 
binations of the ABC class. 

189. In order to letter these combinations we write over 
the first or left-hand file-section the letters AB; next, change 
the letter B into b and write over the second file Ab; next, 
change the letter A and write over the third file aB; next, 
change the letter B and write over the fourth file ab. It will 
be understood that these file letters run all the way down the 



GG 



SIMPLE CATEGORICAL PROPOSITIONS. [ Chap. 7. 



files. Now at the right hand of the upper row of sections 
write C, and at the right hand of the lower row of sections 
write c. 

The row-letters C and c run all the way across, and in what- 
ever sections the file-letters and the row-letters meet we write 
those letters in that section, thus: 

AB Ah aB ab 



ABC 


AbC 


aBC 


abO 


ABc 


Abe 


aEc 


abc 



190. In making an ABC Frame the other way, we make 
a square, then we bisect it by a vertical line; over one section 
we write A and over the other section we write a; then we 
divide the square into four rows by drawing three equi-dis- 
tant horizontal lines, and at the right of the top row we write 
the letters BC ; at the right of the second row we write the let- 
ters Be, at the right of the third row we write the letters bC, 
and at the right of the fourth row we write the letters be. 

191. In this process we have changed a letter at a time, and 
we letter each section with the letters which meet in that sec- 
tion. Our diagram has this appearance: 



ABC 


aBC 


ABc • 


aEc 


AbC 


abC 


Abo 


abc 



Fig. 26. 



BC 
Be 
bC 
be 



192. Of course bv this svstem everv section has a different 



§§ 193-193.] LETTERING THE SECTIONS. G7 

combination. If we change one letter at a time it will be impos- 
sible for us to have two combinations which arc alike. These 
two diagrams are of equal logical value, but I think that the 
first one given explains the theory of logical division more 
clearly than the other, and in practice I have found it more 
convenient. 

193. There is no iron rule about lettering our sections. 
Any plan will do which will give us a different combination for 
each section. Instead of commencing with AB, we might have 
commenced with ab, and then proceeded to change a letter at a 
time. Or, instead of commencing with BC in the second 
example, we might have commenced with be. In fact we can 
commence with any two letters we like, provided that we 
change a letter at a time, and letter each section with the let- 
ters which meet in that section. Neither is it absolutely neces- 
sary that we make a square, a rectangle will do as well. The 
important matter is that we get our eight sections. 

191. Prof. Jevons says that we can read the combinations in 
an ABC Frame by changing the order and by taking either 
one, two, or three terms at a time in two hundred and fifty-five 
different ways, or, in other words, there are two hundred and 
fifty-five imaginable propositions in the Reasoning Frame for 
three terms. In a frame for four terms — A BCD — there are 
sixty-five thousand five hundred and thirty-five conceivable 
propositions, and in a frame for five terms — ABCDE — there are 
four billion two hundred and ninety-four million nine hundred 
and sixty-seven thousand two hundred and ninety-five different 
readings. These different readings may be called alternative 
readings; which of them will eventually remain will depend 
upon the premises. An alternative predicate leaves the sub- 
ject in doubt. It means that the subject is one or more of t lie 
alternatives; this, or the other, or both, or neither, but it does 
not say positively which it is. 

195. It will be noticed in the ABC Frame that there are four 
A's; viz.: ABC, AbC, Abe, ABc, and we can say that A is 
either ABC I ABc I AbC I Abe. Likewise, each of the other 



68 SIMPLE CATEGORICAL PROPOSITIONS. [ Chap. 7. 

letters B, C, a, b, c, lias four combinations, and we can read 
them alternately as we did in the case of A, thus, c is either 
cBA | cbA | cBa | cba. 

196. Again, if we like, we can drop in reading any letter 
and its opposite we choose, thus in the case of c we can drop the 
A's and a's and then our definition is c is either cB | cb. Or, 
again, in reading we can drop the A's, B's, a's and b's and then 
we can say c is c. 

197. We can read the letters in any order we please. If a 
boy has three names, Andrew, Benjamin and Charles, so far as 
logic is concerned it does not make the slightest difference in 
what order we use them. 

198. And again, we can repeat any letter or any combination 
of letters as often as we please, because we know by the Law of 
Identity that A is always A, and AB is always AB, etc., and 
the repetition of a letter or a combination of letters is always 
allowable. 

199. Suppose we get a conclusion which reads: 

ABC. Now, in reading we can drop the B and say A is C, 
and C is A, or we can drop the A and say B is C and C is B. To 
take a concrete example, 

Let A = man, 

B = rational-animal, 

C = reasoning-living-being. 

ABC ean then be read "Man is a rational-animal and a rea- 
soning-living-being," or, we can drop the "rational-animal" and 
say "Man is a reasoning-living-being, and a reasoning-living- 
being is a man." Or we can drop the term "man" and say "A 
rational-animal is a reasoning-living-being, and, a reasoning- 
living-being is a rational-animal." What we have in ABC is 
three names for the one thought; we can use all three of the 
names, or any two of them, or any one of them just as we 
choose. 

200. In the syllogism what is called the "middle term" is 
always dropped in the conclusion. Therefore the syllogism 



201.] 



AN EXAMPLE. 



G'J 



does not give us all the information which it can give. 

201. Let us take the following propositions and ascertain 
what inferences can be deduced from them: 

(1) A man is a rational animal. 

(2) A rational animal is a reasonable living being. 
Let A = man, 

B = rational animal, 
C = reasoning-living-being. 
We can assume 

(3) A rational animal is a man. 

(4) A reasoning-living-being is a rational animal. 
Our premises can then be stated, thus: 

(1) A = B 

(2) B = A 

(3) B = C 

(4) C = B 

Should any one doubt that a rational-animal is a man we can 
prove it by using the Law of the Excluded Middle, thus: A 
rational animal is a man or not-a-man. If a rational animal is 
not-a-man then, since a man is a rational-animal, a man would 
be not-a-man, which is impossible. Therefore, a rational-ani- 
mal is a man. And similarly with the proposition a reasoning- 
living being is a rational animal. 

Xow. make an ABC Reasoning Frame by drawing a rec- 
tangle, then by dividing it into four-files, and then by bisecting 
the four files so as to make two rows. Then letter the files, 
row's and sections, as in the first example: 



AB 


Ab 


aB 


ab 




1 


2 






4 




4 


ABC 


AbC 


aBC 


abC 




1 


2 




3 




3 




A Be 


'In 


HB( 


abc 



Fi< 



70 SIMPLE CATEGORICAL PROPOSITIONS. [ Chap. 7. 

Our first premise is A is B. Of course all the combinations 
of A with b will be inconsistent by the Law of Contradiction, 
because if the proposition A is B is true, the proposition A is b, 
cannot be true at the same time. Therefore, we eliminate the 
combinations AbC and Abe by making a figure 1 in those 
sections. 

Our second premise is that B is A, and therefore, all the com- 
binations of B with a will be inconsistent, because if B is A, B 
is a is contradictory, therefore, we eliminate the combinations 
aBC and aBc by making a figure 2 in those sections. 

Our third premise is that B is C; now any combination of B 
with c will be inconsistent because it implies that B = c, 
hence, we eliminate the combinations ABc and aBc by making 
a figure 3 in those sections. 

Our last premise is that C is B, and of course any combina- 
tion that says that C is b is an inconsistent combination, 
hence, we eliminate the combinations Cba, and CbA, by mak- 
ing a figure 4 in those sections. 

We have now remaining in the Frame two combinations 
ABC and abc. ABC contains our original premises; abc 
automatically remains in the Frame. We can read ABC in 
any order we please and we can use any number of letters we 
choose. We can say A is BC, or AB is C, or BC is A, or CB is A, 
or CA is B, etc. Similarly with abc. To use concrete terms, 
we can say, "What is not-a-man is not-a-rational-animal 
and is not-a-reasoning-living-being," or, we can say, "What is 
not-a-man and what is not-a-rational-animal is not-a-reasoning 
living-being," or, we can say, "Wliat-is-not-a-rational-animal 
and what is not-a-reasoning-liying-being is not-a-man," or, we 
can say, "What is not-a-reasoning-liying-being and what is not- 
a-man is not-a-rational-animal." 

Thus, with one operation we have exhausted all the informa- 
tion contained in our premises and have brought to light a 
great many hidden and latent meanings which were contained 
in them. 

202. Every proposition contains latent meanings, and the 



§§ 203-205.] A ROUTINE PROCESS. 71 

problem of logic is to discover them. The beauty of this sys- 
tem is that it reduces the process of reasoning to a routine pro- 
cess of a few mechanical operations; viz.: the drawing of the 
squares, the lettering of the sections, the stating of the prop- 
ositions and the eliminating of the inconsistent combinations. 
Of course, after we have eliminated all the inconsistent prop 
ositions, the consistent ones must remain. 

203. In eliminating the inconsistent combinations we pay 
attention only to our signs. It is unnecessary to try to realize 
the concrete meaning of our signs during the eliminating pro- 
cess. From the premises in this case, the old logic would 
draw only one conclusion; viz.: A is C, "A man is a reasoning- 
living-being." So that in its ability to draw every possible 
conclusion which can be drawn from the premises by one oper- 
ation, our system is greatly superior to the old logic. 

204. From the premises A is B, B is A, B is C and C is B, we 
derive the conclusions which can be read in the combination 
abc. By the old logic this process is called deduction, and it 
is said to be a very different process from induction. 

205. In induction the problem is — given the conclusions to 
obtain the premises. Now let us take some of the conclusions 
contained in abc and use them as premises and ascertain what 
the result will be by our system. Let us take these conclusions: 

(1) "What is not-a-man is not-a-rational-animal." 

(2) "What is not-a-rational-animal is not-a-reasoning-living- 
leing." 

We can read these propositions backward, because the pred- 
icates are synonymous with the subject, and thus obtain two 
more conclusions; viz.: 

(3) "What is not-a-rational-animal is not-a-man." 

(4) "What is not-a-reasoning-living-being is not-a-rational- 
animal." 

It is always necessary to make our propositions read back- 
ward as well as forward. 
Let a == not-man, 



72 SIMPLE CATEGORICAL PROPOSITIONS. [ Chap. 7. 

b = not-rational-animal, 
c = not-reasoning-living-being. 
Our premises can be stated symbolically thus: 

(1) a = b 

(2) b = a 

(3) b = o 

(4) c = b 

Make an ABC diagram and letter it, thus : 



AB 


Ab 


aB 


ab 




2 


1 






3 




3 




2 


1 




4 




4 





Fig. 28. 

Now, as a is b, then every combination containing aB will be 
inconsistent by the Law of Contradiction; therefore, the com- 
binations aBc, and aBC, are inconsistent with premise No. 1, 
and we make a figure 1 in those sections, which indicates that 
they are inconsistent with premise No. 1. 

We put figures in the sections eliminated to show which 
premises those sections are inconsistent with. 

Now, as b is a, the combinations containing bA will be incon- 
sistent by the Law of Contradiction, therefore, the combina- 
tions bAC and bAc are inconsistent and we eliminate them by 
making the figure 2 in those sections. 

And, again, as b is c, any combination having bC, will be 
inconsistent, therefore, the combinations bCA and bCa are 
inconsistent and we eliminate them by making a figure 3 in 
those sections. 

And, lastly, c is b. This means that c can combine with b 
only, and that any combination of c with B is inconsistent, 
therefore, the combinations cBA and cBa are inconsistent and 



§§ 206-208.] NEGATIVE PROPOSITIONS. 73 

we eliminate them by making a figure 4 in those sections. 

The following combinations automatically remain in the 
diagram; viz: abc and ABC; abc is our premises, and from 
these premises we derive the conclusions contained in the com- 
bination ABC. 

From among the various readings of this combination we 
can take the following: 

(1) A is B 

(2) B is C 

(3) A is C 

which we can translate thus: 

(1) "A man is a rational animal." 

(2) "A rational animal is a reasoning living being." 

(3) "A man is a reasoning living being." 

206. We thus see that our system works equally well back- 
ward and forward, and that, so to speak, it proves itself. 
From the original premises in ABC we derive the conclusions 
in abc, and from the conclusions in abc we derive the original 
premises in ABC. It also proves that deduction and induc- 
tion are merely reverse operations, and that in a true logic we 
can reason from the conclusions to the premises just as easily 
and just as accurately as we can reason from premises to con- 
clusions. 

207. According to the old logic propositions containing 
negative terms were called, incorrectly, negative propositions, 
and it was a rule of the old logic that from two negative prop,: 
ositions no conclusions could be deduced. According to the 
old logic the propositions, "What is not a man is not a rational 
animal," and "What is not a rational animal is not a reasoning 
living being," would be negative propositions, and, therefore, 
no conclusions could be deduced from them. But the example 
which we have just worked demonstrates that this is another 
of the mistakes, as I think, of the old logic. 

208. Let us take these propositions: 

(1) "What is not fit to learn is not proper to teach." 



74 



SIMPLE CATEGORICAL PROPOSITIONS. [ Chap. 7. 



(2) "What is not proper to teach ought not to be printed," 
and ascertain what conclusions we can derive from them. As 
these are synonymous propositions we can read them back- 
ward and thus get two more propositions, viz.: 

(3) "What is not proper to teach is not fit to learn. ,, 

(4) "What ought not to be printed is not proper to teach." 
Let a = what is not-fit-to-learn, 

b = what is not-proper-to-teach, 

c = what ought not-to-be-printed. 
Always let negative signs stand for negative terms. Our 
propositions can be stated, thus: 

(1) a = b 

(2) b = c 

(3) b = a 

(4) c = b 

Make an ABC Reasoning Frame by dividing a square into 
four files and the four files into two rows, and then letter the 
files, rows and sections, thus: 



AB 


Ab 


aB 


ab 




2 


1 


2 




3 






\BC 


AbC 


aBC 


abC 




3 


1 




4 




4 




A Be 


Abe 


aBc 


abc 



Fig. 29. 



Then, as a is b.any combination of a with B will be inconsist- 
ent. We therefore eliminate the combinations aBC and aBc 
by making a figure 1 in those sections. 

Next, as b is c, every combination containing b and C will 
be inconsistent and we eliminate it by making a figure 2 in 
those sections. This will cause us to eliminate the sections 
AbC and abC. 

Again, as b is a, any combination of b with A will be incon- 



§ 209.] NEGATIVE PROPOSITIONS. 75 

sistent. The combinations AbC and Abe are therefore incon- 
sistent and we eliminate them by making a figure 3 in those 
sections. 

And, lastly, c is b. Any combinations, therefore, which say 
that c is B are inconsistent and we eliminate them by making 
a figure 4 in those sections. The combinations ABc and aBc 
are inconsistent. The combinations abc and ABC automat- 
ically remain in the Reasoning Frame. The combination abc 
contains the premises, and the combination ABC contains the 
conclusions. 

We can read ABC in a variety of ways; among others, we 
can read A is B, B is C, and A is C, thus: 

(1) "What is fit to learn is proper to teach." 

(2) "What is proper to teach ought to be printed," and 

(3) "What is fit to learn ought to be printed." 

By a previous example we have already learned that if we 
take these conclusions, A is B, and B is C, for premises, we 
shall get as conclusions the inferences contained in abc. 

209. Let us take these propositions (from Jevons) : 

(1) London is the capital of England. 

(2) London is the most populous city in the world. 

We can read these propositions backward and obtain two 
more propositions. 

(3) The capital of England is London. 

(4) The most populous city in the world is London. 

We should always bear in mind that it is necessary to state 
all the prima facie meanings of our propositions. 

Let A = London, 

B = the capital of England, 

C = the most populous city in the world. 

Our premises can then be stated, tlrus: 

(1) A = B 

(2) A = C 

(3) B = A 

(4) G = A 



76 



SIMPLE CATEGORICAL PROPOSITIONS. [ Chap 7. 



Next make an ABC diagram and letter it as hereinbefore 
described. 



AB 


Ab, 


aR 


aR 






1 


3 
4 


4 


C 


2 





3 




c 



Fig. 30. 

Now, as A is B, any combination of A with b, which means 
A is b, is an inconsistent combination. The combinations AbC 
and Abe are inconsistent and we eliminate them by making a 
figure 1 in those sections. 

Again, as A is C, all the combinations containing Ac are 
inconsistent combinations, and we eliminate them by making 
a figure 2 in those sections. The combinations ABc and Abe 
are inconsistent. 

Again, as B is A, the combinations containing Ba are incon- 
sistent, and we eliminate them by making a figure 3 in those 
sections. The combinations aBC and aBc are inconsistent. 

And, lastly, as C is A, the combinations containing Ca are 
inconsistent, and we eliminate them by making a figure 4 in 
those sections. The combinations aBC and abC are incon- 
sistent. 

The following combinations automatically remain, viz.: ABC 
and abc. The combination ABC contains our premises; the 
combination abc contains our conclusions. The conclusion 
which the old system would draw is, "The capital of England 
is the most populous city in the world." This is contained in 
the combination ABC. 

But besides the conclusion which the old logic would draw 
we have all the conclusions contained in the combination abc, 
and we can read, 

(1) "What is not London is not the capital of England," 



§ 21 \] 



EXAMPLES. 



77 



(2) "What is not London is not the most populous city in 
the world." 

(3) "What is not the capital of England is not the most 
populous city in the world." 

(4) "What is not the most populous city in the world is not 
London and is not the capital of England," etc. 

210. Let us take these propositions (from Jevons): 

(1) The substance of least density is hydrogen. 

(2) The substance of least atomic weight is hydrogen. 

As these are synonymous propositions we can read them 
backward, and obtain two more propositions, thus: 

(3) Hydrogen is substance of least density. 

(4) Hydrogen is substance of least atomic weight. 
Let A = substance of least density, 

B = substance of least atomic weight, 
C = hydrogen. 

Our propositions can be stated, thus: 

(1) A = C 

(2) B = C 

(3) C = A 

(4) C = B 

Make an ABC diagram and letter it as usual, thus: 



AB 


Ab 


aB 
3 


ab 

3 
4 


C 




4 


1 
2 


1 


2 




c 



Fig. 31. 



The following combinations are inconsistent with A is and 
we eliminate them by making a figure 1 in those sections; viz: 
ABc, Abe. 



78 SIMPLE CATEGORICAL PROPOSITIONS. [ Cliap. 7. 

The following combinations are inconsistent with B is C, 
and we eliminate them by making a figure 2 in those sections; 
viz.: ABc and aBc. 

The folowing combinations are inconsistent with C is A, 
and we eliminate them by making a figure 3 in those sections; 
viz. : aBC and abC. 

Lastly, as C is B, every C which is combined with b is incon- 
sistent and we eliminate it by making a figure' 4 in those sec- 
tions. The combinations AbC and abC are inconsistent. 

The following combinations remain: ABC and abc. In ABC 
we have our original premises, and of course we have several 
other readings besides; for instance, we can read A is B, B is 
A, and translate thus: 

(1) "The substance of least density is the substance of least 
atomic weight.'' 

(2) The substance of least atomic weight is the substance 
of least density." 

In abc we can read, 

(1) "What is not the substance of least density is not the 
substance of least atomic weight." 

(2) "What is not the substance of least atomic weight is 
not hydrogen." 

(3) "What is not hydrogen is not the substance of least 
density, etc. 

211. Let us take two of the conclusions obtained in the last 
example for premises. 

(1) What is not the substance of least density is not the 
substance of least atomic weight, 

(2) What is not the substance of least atomic weight is not 
hydrogen. 

Let a = what is not substance of least density, 

b = what is not substance of least atomic weight, 
c = not hydrogen. 

212. As the subject and predicate are synonymous terms 
we can read the propositions backward and get two additional 
premises. The reason why we wish to get additional premises 



§ 213.] 



FROM CONCLUSIONS TO PREMISES. 



?9 



is this: In a Reasoning Frame the various combinations repre- 
sent alternative propositions, and every premise given us 
enables us to strike out some of these alternative propositions, 
and the more alternative propositions we can strike out, the 
more definite will be the propositions which remain. 

213. A proposition in the alternative does not give us that 
definite information of a subject which we desire. Our object 
is to get rid of as many alternatives as we can, so that the 
information which remains in the Frame shall be positive and 
definite, and every additional premise which enables us to 
strike out one or more of our alternative combinations is a 
help to this end. 

Our premises can be stated symbolically as follows: 

(1) a = b 

(2) b = c 

(3) b = a 

(4) c = b. 

Now, if a is b, then every combination of aB, which means 
that a is B, is inconsistent by the Law of Contradiction. The 
combinations aBC and aBc are inconsistent and we eliminate 
them by making a figure 1 in those sections: 

AB Ab aB ab 



2 1 2 
3 

3 1 

4 4 



Fig. 32. 

And, if b is c, then every combination of bC is inconsistent. 
The combinations AbC and abC are inconsistent and we elim- 
inate them by making a figure 2 in those sections. 

And if b is a, then the combinations bA are inconsistent and 
we eliminate them by making a figure 3 in the sections marked 
AbC and Abe. 



80 SIMPLE CATEGORICAL PROPOSITIONS. [ Chap. 7. 

And if c is b, then the combinations ABc and aBc are incon- 
sistent and we eliminate them by making a figure 4 in those 
sections. 

The combinations abc and ABO automatically remain in the 
Frame; abc contains the premises and ABC contains the con- 
clusions. In the combination abc we can find readings which 
were not stated in our premises, and these readings can pro- 
perly be called conclusions. For instance, we can read a is c 
and c is a, which being translated read, 

(1) "What is not-the-substance-of-least-density is not-hydrc- 
gen." • 

(2) "What is not-hydrogen is not-the-substance-of-least-den- 
sity." 

In addition to these conclusions, we have also the conclusions 
contained in the combination ABC, and from that combination 
we can read, 

(1) "Hydrogen is the substance of least density." 

(2) "Hydrogen is the substance of least atomic weight." 

(3) "The substance of least atomic weight is the substance 
of least density," etc. 

This example demonstrates again the fact that we can rea- 
son from premises to conclusions and from conclusions to 
premises by our system with equal ease and accuracy. 

214. Let us now take some inconsistent propositions and 
ascertain what the result will be. 

(1) The Queen of England is not the Empress of India, 

(2) Queen Victoria is the Queen of England. 

(3) The Empress of India is Queen Victoria, 

Let A = Queen of England, 

h = not-the-Empress of India, 
C = Queen Victoria, 
B = Empress of India, 

The second and third propositions are synonymous and can 
be read backward; thus giving us two more propositions; vi*;: 

(4) The Queen of England is Queen Victoria. 



§ 174.]- 



INCONSISTENT PROPOSITIONS. 



81 



(5) Queen Victoria is the Empress of India. 

174. Make a square and divide it into eight sections and let- 



ter them as heretofore: 



AB Ab aB ab 



1 




2 


2 


ABC 


AbC 


aBC 


abC 


1 
3 
ABc 


Abe 


3 
aBc 


abc 



Fig. 33. 
Our premises can be stated as follows: 



(1) A = Ab 

(2) = A 

(3) B = C 

(4) A = C 

(5) C = B 

Now, if A is Ab, then the combinations which contain AB 
are inconsistent. The combinations ABC and ABc are incon- 
sistent and we eliminate them by making a figure 1 in those 
sections. 

And if G is A, then the combinations containing Ca are 
inconsistent. We therefore eliminate the combinations aBC 
and abC by making a figure 2 in those sections. 

Again, if B is C, then the combinations containing Be are 
inconsistent. We therefore eliminate the combinations ABo 
and aBc by making a figure 3 in those sections. 

By looking at our diagram, we now see that the following 
combinations have been eliminated, viz: ABC, aBC, ABc ant/ 
aBc. There are no other B's in the Frame; all the B's having 
been eliminated, the diagram tells us that our premises are 
inconsistent and that it is of no use for us to eliminate any more 
combinations. The diagram says that our propositions have 
6 



82 SIMPLE CATEGORICAL PROPOSITIONS. [Chap. 7. 

affirmed that there was an Empress of India, and there was not 
an Empress of India. 

215. Having discovered an inconsistency in our premises, it 
is of no use for us to proceed in the working of the problem set 
before us, viz. : to find the conclusions which could be deduced 
from the given premises. 

216. The Law of Opposites told us that we could not have 
an idea without its opposite; that if we have the idea of 
straightness, for instance, we must have the idea of not- 
straightness; if we have the idea of light, we must have the 
idea of not-light, and that if we posit one, we posit the other. 
And, consequently, if we eliminate one, we must eliminate the 
other. Now, as we had no B's left in our Reasoning Frame, 
we must eliminate all the b's, because we cannot have one 
without the other. If we eliminate the b combinations which 
are left in the Frame, viz.: AbC, Abe and abc, there is nothing 
left in the Frame for us to reason about. 

And, again, as C = B by our premises, if there is not any B, 
then there can not be any C. And, as A = C by our premises, 
if there is not any C, then there can not be any A. 

217. Whenever the premises are consistent and our elimina- 
tions have been correctly performed, every letter both positive 
and negative, will remain in the Reasoning Frame and appear 
in the conclusions. The absence of a single letter tells us that 
our premises are inconsistent. 

218. Let us take another example of inconsistent proposi- 
tions: 

(1) Grover Cleveland is President of the United States. 

(2) The President of the United States is Commander in 
Chief of the Army of the United States. 

(3) The Commander in Chief of the Army of the United 
States is not Grover Cleveland. 

In (1) and (2) the subjects and predicates are synonymous 
and we can read them backward and get two additional pro- 
positions, viz. : 

(4) The President of the United States is Grover Cleveland. 



§218.] 



INCONSISTENT PROPOSITIONS. 



83 



(5) The Commander in Chief of the Army of the United 
States is the President of the United States. 

We can make (3) a synonymous proposition by adding the 
subject to the predicate, and it will then read "The Commander 
in Chief of the Army of the United States is the Commander in 
Chief of the Army of the United States not Grover Cleveland." 
In doing this we shall have to disregard the laws of rhetoric, 
but, as we have said before, the science of logic is independent 
of other sciences, and in solving its problems it must be 
allowed to make its own rules and have its own methods. 
Having converted (3) into a synonymous proposition, we can 
read it backward. 

Let A = Grover Cleveland, 

B = The President of the United States, 

C = Commander in Chief of the Army of the United 

States, 
a = not-Grover Cleveland. 
Our premises can be stated as follows: 

(1) A = B 

(2) B = 

(3) C = aC 

(4) B = A 

(5) C == B 

Draw a rectangle and divide it into eight sections and letter 
them as heretofore: 



AB 


Ab 


aB 


ab 


3 

ABC 


1 

3 

AbC 


aBC 


abC 


2 


1 


2 




ABc 


Abe 


aBc 


abc 



Fiff. 34. 



Now, if A is B, then the combinations which contain Ab are 
inconsistent. To put it in other words, wherever we hai 



84 SIMPLE CATEGORICAL PROPOSITIONS. [ Chap. 7. 

A, we must find B with it; if we find an A with b we must 
eliminate the combinations. We therefore eliminate the com- 
binations AbC, and Abe by making a figure 1 in those sections. 

Again, if B is C, then every combination Be is inconsistent. 
We therefore eliminate the combinations ABc and aBc by mak- 
ing a figure 2 in those sections. 

Again, if C is aC, then any combination containing AC is an 
inconsistent combination. We therefore eliminate the com- 
binations ABC and AbC by making a figure 3 in those sections. 

Now, all our A's have disappeared from the Reasoning 
Frame. This tells us that our premises are inconsistent and it 
is useless for us to proceed. 

EXAMPLES FOR PRACTICE. 

219. What inferences can be deduced from the following 
pairs of premises: 

(1) aB = aBC 
ab = abc 

(2) aB = aBo 
ab = abC 

(3) A=ABC 
c = cab 

(4) C = CA 
c = cab 

(5) C = Ca 

c = cAB 

( 6) aB = aBC 
A = Abo 



CHAPTER VIII. 

INDUCTION. 

220. We have already learned that in logic a proposition is 
two or more names, titles, designations or descriptions of the 
same idea, and that we can represent these names by letters. 
And that a square represents our Universe of Discourse or 
Field of Thought, and that the combinations contained in the 
squares represent all the possible propositions which can be 
made from the terms of our premises. 

221. In inductive logic, so called, the problem is to find 
premises which will produce the conclusions which are given 
to us. In the last chapter we have seen that, given the prem- 
ises contained in ABO we can get the conclusions contained in 
abc; and given the premises in abc we can get the conclusions 
contained in ABC. 

222. The problem which we are now to consider is — given 
as conclusions ABC and abc, what premises will produce 
all the conclusions contained in ABC and abc. From ABC we 
can get propositions reading A is B, A is C, etc. These pro- 
positions are called definitions. A is B is a. definition of A, B 
is C is a definition of B, b is c is a definition of b. A complete 
definition of A in ABC would be, A is BC, and a complete 
definition of a in abc is, a is be. 

223. We can always define a letter-term contained in only 
one conclusion by saying that it is the other letters contaiued 
in the conclusion. 

Now, if A is BC, it is clear that all the other combinations 
of A, viz. : ABc, AbC and Abc, are inconsistent combinations. 
If A is BC, there is no other definition of A in the ABC 
Reasoning Frame. This is just as true as the axiom that a 
thing cannot be in two places at the same time. 

Similarly, if the definition of a is, a is be, then the other com- 



86 INDUCTION. [Chap. 8. 

binations of a are inconsistent. Thus: aBC, aBc and abC are 
inconsistent. To illustrate this concretely, let us suppose that 

A = man, 

B = rational-animal, 

— - reasoning-living-being, 

a = not-man, 

b = not-rational-animal, 

c = not-reasoning-living-being. 

With these terms we can make a number of propositions which 
can be stated symbolically, thus: 

(1) A=:B 

(2) U=C 

(3) A = 

(4) b = a 

(5) c = b 

(6) c = a 
These can be translated as follows: 

(1) Man is a rational animal. 

(2) A rational animal is a living reasoning being, 

(3) Man is a reasoning living being. 

(4) What is not-a-rational-animal is not-a-man, 

(5) What is not-a-reasoning-living-being is not-a-rational- 
animal. 

(6) What is not-a-reasoning-living-being is not-a-man. 

Or course, we could get several other propositions from these 
pr« raises, but the above are sufficient. The problem is, What 
picmises will produce all these conclusions? Now, from these 
conclusions let us get a definition of the positive term man and 
of its nf gative not-man. A complete definition of the positive 
term man will be; A man is a rational animal and a reasoning 
living being. And a complete definition of the term not-man, 
is What is not-a-man is not-a-rational-animal and not-a-reason- 
ing-living-being. We can state them symbolically thus: 

(1) A = BC 

(2) a == be 



§ 2£8.] 



EXAMPLES. 



87 



Make a rectangle, dividing it into eight sections, and letter as 
heretofore : 

AB Ab aB ab 





1 


2 


2 


ABC 


AbC 


aBC 


abC 


1 


1 


2 




ABc 


Abe 


aBc 


abc 



Fig. 35. 

Now, if A is BC, then the combinations ABc, AbC, Abc, are 
inconsistent and we eliminate them by making a figure 1 in 
those sections. 

Again, if a is be, then the combinations abC, aBC and aBc, 
are inconsistent, and we eliminate them by making a figure 
2 in those sections. 

The conclusions contained in ABC and abc automatically 
remain in the Frame. And, thus, from the definitions of A and 
a we have obtained all the conclusions which are contained in 
those combinations. Again, if we take these definitions, 

(1) A rational animal is a man and a reasoning living being, 

(2) What is not-a-rational-animal is not-a-man and is not-a- 
reasoning-living-being, 

we shr.li get the same results. We can state these definitions 
symbolically, thus: 

(1) B = AC 

(2) b =-• ac 

Make a rectangle, divide it into eight sections and letter as 
heretofore: 

AB Ab aB ab 





2 


1 


2 


ABC 


AbC 


aBC 


abC 


1 


2 


1 




ABc 


Abc 


aBc 


abc 



Fig. 3G. 



88 



INDUCTION. 



[Chap. 8. 



Jf l.» is AC, then the combinations aBc, BAc, and aBO are 
inconsistent and we eliminate them by making a figure 1 in 
those sections. 

And if b is ac, then the combinations abC, AbC and Abe are 
inconsistent and we eliminate them by making a figure 2 in 
those sections. 

The combinations ABC and abc automatically remain in the 
Frame, and thus from the complete definition of the positive 
term, "rational animal," and the complete definition of its neg- 
ative, "not-rational-animal," we have obtained all the conclu- 
sions which can be read in the combinations ABC and abc. 

224. From this we can deduce the general rule: That the 
complete definitions of a positive term and its negative which 
can be obtained from the conclusions, will give us all the 
premises. 

225. There are other ways of obtaining the premises for 
given conclusions, but the above is the simplest and easiest 
method. Another way to obtain the premises for given conclu- 
sions is, when the conclusions are categoricals, to get a defini- 
tion of every one of the positive letters, or a definition of every 
one of the negative letters, and then to use one or the other 
set of definitions for premises. Thus from the conclusion 
ABC we can get the definitions: 

(1) A = ABO 

(2) B = BAC 

(3) C = CAB 

Make an ABC diagram and letter it as heretofore: 

AB Ab aB ab 





1 


2 


3 




3 


3 




ABC 


AbC 


aBC 


abC 


1 


1 


2 




2 








ABc 


Abc 


aBc 


abc 



Fig. 37. 



§ 226.] 



PREMISES FROM CONCLUSIONS. 



89 



Now if A is BC we must eliminate the combinations AbC, 
ABc, Abe. Make a figure 1 in those sections. 

Now, if B is AC, we must eliminate the combinations ABc, 
aBC and aBc. Make a figure 2 in those sections. 

Now, if C is AB, then the combinations AbC, abC, aBC are 
inconsistent and we eliminate them by making a figure 3 in 
those sections. 

The combinations ABC and abc automatically remain. 
Thus, from the complete definitions of all the positive terms we 
have all the conclusions which can be read in the combinations 
ABC and abc. 

226. The reader will notice that, in this case, we have not 
been reading our propositions backward. The reason for our 
not doing so is because it would be a useless proceeding. That 
is, by reading these definitions backward, we cannot eliminate 
any more propositions than we can by simply reading them for- 
ward. Let us illustrate this rule further by taking definitions 
of the negative terms. 
Let (1) a = abo 

(2) b = bac 

(3) c = cab 

JIake an ABC diagram and letter it as heretofore: 

AB Ab aB ab 





2 


1 


1 
2 


ABC 


AbC 


aBC 


abC 


3 


2 


1 






3 


3 




ABc 


Abc 


aBc 


abc 



Fig. 38. 



90 INDUCTION. [ Chap. 8. 

Now, if a is be, then the combinations abC, aBC, aBc, are 
inconsistent, and we eliminate them by making a figure 1 in 
those sections. 

Again, if b is ac, then the combinations abC, AbC, Abe, are 
inconsistent, and we eliminate them by making a figure 2 in 
those sections. 

Again, if c is ab, then the combinations ABc, Abe, aBc are 
inconsistent, and we eliminate them by making a figure 3 in 
those sections. 

The combinations ABC and abc automatically remain, and 
thus from the complete definitions of the negative terms used 
in the given conclusions, we can get premises which will pro- 
duce the given conclusoins. 

227. The old logic did not profess to have any method for 
obtaining the premises for given conclusions. So far as I 
know, Prof. Jevons was the first logician to undertake to solve 
the problem. Ordinarily, he is a very clear writer on the old 
logic, and his text book on logic is deservedly popular. But in 
treating of Induction he has made mistakes. We have seen 
that the Inductive problem is quite as easy of solution as the 
Deductive problem. But in Jevon's Principles of Science, p. 
121, we read, "It must be allowed that Inductive investigations 
are of a higher degree of difficulty and complexity than any 
questions of Deduction; and it is this fact, no doubt, which led 
some logicians, such as Francis Bacon, Locke, and J. S. Mill to 
erroneous opinions concerning the exclusive importance of 
Induction. In Induction all is inverted. The truths to be 
ascertained are more general than the data from which they are 
drawn. The process by which they are reached is analytical 
and consists in separating the complex combinations in which 
natural phenomena are presented to us and determining the 
relations of separate qualities. 

Given events obeying certain unknown laws, we have to dis- 
cover the laws obeyed. Instead of the comparatively easy task 
of finding what effects will follow from a given law, the effects 
are now given and the law is required." 



§§ 228-231.] JEVONS ON INDUCTION. 91 

228. The reader will have observed that in our system it 
is quite easy to state our conclusions, i. e., effects, symboli- 
cally and from those symbolical statements of conclusions, 
L e. y effects, to obtain definitions either of a positive term and 
its negative, or of each of the positive terms or of each of the 
negative terms, and by using these definitions as premises to 
obtain the given conclusions, i. e., effects. 

229. Again, Prof. Jevons says, "Differentiation, the direct 
process (£. e., Deduction), is always capable of performance by 
fixed rules, but as these rules produce considerable variety of 
results, the inverse process of Integration (i. c, Induction), pre- 
sents immense difficulties, and in an infinite majority of cases 
surpasses the present resources of mathematicians. There are 
no infallible and general rules for its accomplishment and it 
must be done by trial, guess-w T ork, or by remembering the 
re ults of Differentiation and using them as a guide." The 
Inverse process does not present "immense difficulties." It is 
quite as easy, as we have shown, as Deduction. The rules for 
finding premises for conclusions are as "general and infallible" 
as the rules for finding conclusions from premises. Guess- 
work has no place in any logical system. 

230. On p. 125, Principles of Science, he says to the reader, 
"To test the facility with which he can solve this Inductive 
problem, let him casually strike out any of the combinations of 
the fourth column of the logical alphabet and say what laws the 
remaining combinations obey. Observing that every one of 
the letter-terms and their negatives ought tc appear, in order 
to avoid self contradiction in the premises." 

231. We have not reached in this work combinations con- 
taining four terms, but we can try the experiment equally well 
with combinations involving three terms. Let us make an 
ABC diagram, as usual, thus: 



02 



INDUCTION. 
AB Ab aB ab 



X 




X 


X 


ABC 


AbC 


aBC 


abC 




X 




X 


ABc 


Abe 


aBc 


abc 



[ Chap. «. 



Fig. 39. 

And let us casually strike out these five combinations, ABC, 
Abc, aBC, abC, abc. Make an X in those sections. The com 
binations which remain are ABc, AbC, aBc. These represent 
the given conclusions. The problem is. What premises will 
produce these conclusions? This is a very easy task. All we 
have to do is to obtain from these three conclusions definitions 
of A and a, or of B and b, or of C and c, and any one of these 
pairs will furnish the required answer. 

The reader will notice that there are two combinations in 
which A occurs, viz.: ABc and AbC. In this case we cannot 
say that A is either one of these combinations alone, but it is 
one or the other. The definition of A, therefore, is, 
A = ABc | AbC 

There is only one combination containing a; the definition of 
a, therefore, is, 
a = aBc 

Now 7 , if A is either ABc or AbC, then the combinations ABC 
and Abc are inconsistent and we eliminate them. 

And if a is aBc, then the combinations aBC, abC, abc, are 
inconsistent and we eliminate them. The original conclusions 
ABc, AbC and aBc automatically remain. 

Similarly, if we take the definitions of B and b. The defini- 
tion of B is, 

B = ABc | aBc 

The definition of b is, 
b = AbC 



§ 232.] 



PREMISES FROM CONCLUSIONS. 



93 



Now, if B is either ABc or aBc, then the combinations ABC 
and aBO are inconsistent and we eliminate them. 

And if b is AbC, then the combinations Abe, abC and abc are 
inconsistent and we eliminate them. 

And again we have the given conclusions, ABc, AbC and 
aBc. 

232. We have seen early in this chapter that where the 
given conclusions were ABC and abc, that by getting defini- 
tions of each of the positive terms, or of each of the negative 
terms, and by using either set of definitions for premises, we 
could obtain the given conclusions. This method succeeds 
where we have no alternative definitions. But when we have 
alternative definitions in the given conclusions, then we must 
either use the process of taking the definitions of a letter and its 
negative or we must, so to speak, feel our way in getting the 
required premises. Thus: the definition of a in the case before 
us is, 

a = aBc 
b = AbC 

Make an ABC diagram and letter it as usual, thus: 

AB Ab aB ab 



3 




1 

3 


2 
3 




2 




1 
2 



Fig. 40. 



Hereafter it will be unnecessary to letter the sections, 
because, by this time, the reader will understand that each sec- 
tion is supposed to be lettered with the letters which would 
meet in that section. 

Now, if a is aBc, then the combinations aBC, abC and abc 
are inconsistent and we eliminate them by making a figure 1 in 
those sections. 



94 



INDUCTION. 



[Chap. 8. 



Again, if b is AbC, then the combinations abC, abc, Abe, are 
inconsistent, and we eliminate them by making a figure 2 in 
those sections. 

But now the reader will perceive that a definition of c would 
not cause us to strike out the combination ABC. The defini- 
tion of c is, 

c = ABc | aBc 
and the only combinations which are inconsistent with this 
definition are Abc and abc, and these have already been elimi- 
nated. 

If, however, we take the definition of C, which is, 
C = AbC 
then the combinations ABC, aBC, abC, are inconsistent, and we 
eliminate them by making a figure 3 in those sections. So 
that the definitions of a, and b, and C are the premises required. 

This tentative method of finding premises for given conclu- 
sions can be followed in all cases. 

233. Let us make another ABC diagram and letter it as 
usual, thus: 

AB Ab aB ab 





X 




X 




■) 








1 




1 




3 




5 


X 




X 




4 








2 




a 




3 




4 





Fig. 41. 



Now let us strike out four sections, viz.: ABc, AbC, aBc and 
abC. In this case the definitions of C and c are, 
C = ABC | aBC 
c = Abc | abc 
If C is ABC or aBC, then the combinations AbC and abC are 
inconsistent and we eliminate them by making a figure 1 in 
those sections. 



£§ 233, 234.] 



PREMISES FROM CONCLUSIONS. 



95 



Again, if c is Abe or abc, then the combinations ABc and aBc 
are inconsistent and we eliminate them by making a figure 2 in 
those sections. 

Thus, from these premises we have obtained the given conclu- 
sions ABC, aBC, Abc, abc. 

Let us try the tentative method of obtaining definitions with 
this problem. 

The definition of A is, 
A = ABC | Abc 

Then the combinations AbC and ABc are inconsistent and we 
eliminate them by making a figure 3 in those sections. 

The definition of B is, 
B == ABC | aBC 

Then the combinations ABc and aBc are inconsistent and we 
eliminate them by making a figure 4 in those sections. 

The definition of C is, 
C — ABC | aBC 

Then the combinations AbC and abC are inconsistent, and 
we eliminate them by making a figure 5 in those sections. 

The given conclusions remain. Thus we see that in this case 
the definitions of ABC, which we obtained from the given con- 
clusions, have furnished us the required premises. 

234. Let us make another ABC diagram and letter it as 
usual and strike out three combinations, thus: 



AB 


Ab 


aB 


ab 




X 


X 






2 


1 






• 


X 

1 





Fig. 42. 



Let us strike out AbC, aBc, aBC. The definidons of B and b 
are, 



t6 



INDUCTION. 



[ Chap. 8. 



B = ABC | ABc 

b = Abe | abc | abC 

If B is ABC or ABc. then the combinations aBC and aBc are 
inconsistent, and we eliminate them by making a figure 1 in 
those sections. 

And if b is either Abc or abC or abc, then the combination 
AbC is inconsistent and we eliminate it by making a figure 2 in 
that section. The sections which remain without a figure are 
the given conclusions. And the definitions of B and b have 
been the premises which have yielded the given conclusions. 

235. We obtained our given conclusions in the first place by 
striking out casually three combinations, viz.: AbC, aBC, 
aBc. After striking out these combinations, the combinations 
which remained represented the given conclusions for which 
we were to find the premises. Our rule told us that the defi- 
nitions of a term and its negative will always yield us premises 
which will produce the given conclusions. We try this process 
and we find that it causes us, in every case, to strike out again 
tin- same combinations which we casually struck out in the 
tiisi place. We indicate the re-striking out process by the fig- 
ures which we put in the sections. The conclusions which 
remain are exactly the same as those obtained by the casual 
striking-out process. 

Casually striking out, means striking out by chance or with- 
out design. 

236. Let us make another ABC diagram and strike out, 
casually, two combinations. 

AB Ab aB ab 



X 

1 










X 

•> 







Fig. 43. 



§ 237.] AN EXAMPLE. 97 

We will strike out ABC and Abe. The conclusions which 
remain are AbC, aBC, abC, ABc, aBc, abc. 

The definitions of B and b are, 

B = ABc I aBC | aBo 
b = AbC I abC | abc 

If B = ABc I aBC | aBc, then ABC is inconsistent, and we 
eliminate it by making a figure 1 in that section. 

If b is AbC or abC or abc, then Abc is inconsistent, and we 
eliminate it by making a figure 2 in that section. 

Thus we see that the definitions of B and b which we obtained 
from the given conclusions caused us to strike out exactly the 
same combinations which we struck out casually in order to 
obtain the given conclusions. 

We could just as well have taken the definitions of A and a, 
or of C and c, and either of them would have produced the 
given conclusions. 

237. Let us now take a concrete example. Suppose that 
these propositions are given to us as conclusions, and we are 
required to find what premises will produce them. 

(1) The powers delegated to the United States by the Con- 

stitution are not reserved to the States and are not 
reserved to the people. 

(2) The powers reserved to the States are not reserved to 

the people. 

(3) The powers not reserved to the States and not reserved 

to the people are delegated to the United States. 

(1) Let A = powers delegated to the United States, 
B = the powers reserved to the States, 
C = the powers reserved to the people, 
a = the powers not-delegated-to-the-United States, 
b = the powers not-reserved-to-the-States, 
c = the powers not-reserved-to-the-people. 

The propositions can be stated, thus: 

(1) A = Abc 

(2) B = Be 

(3) be = Abc 
7 



98 



INDUCTION. 



[ Chap. 8. 



Then make an ABC diagram, and write down the file-letters 
and the row-letters. 

Remember that a figure in a section indicates that the com- 
bination in that section is inconsistent with the proposition 
having the same number. 

AB Ab aB ab 



1 


1 


• ) 




1 






a 



Fig. 44. 



Now, if A is Abe, then the combinations ABC, ABc, AbC, are 
inconsistent, and we eliminate them by making a figure 1 in 
those sections. 

Again, if B is Be, then the combinations ABC and aBC are 
inconsistent, and we eliminate them by making a figure 2 in 
those sections. 

Again, if be is Abe, then the combination abc is inconsistent, 
and we eliminate it by making a figure 3 in that section. 

The following combinations remain as consistent proposi- 
tions, viz.: Abc, aBc, abC, and from them we can get the fol- 
lowing definitions: 

(1) A = Abc 

(2) B = Bac 

(3) C = Cab 

(4) a = aBc | abC 

(5) b = baC | bAc 
(G) c = cAb | caB 

These propositions may be translated into concrete terms, as 
follows: 

(1) The powers delegated to the United States are not 
reserved to .the States and are not reserved to the 
people. 



§§ 238, 239.] AN EXAMPLE. 99 

(2) The powers reserved to the States are not delegated ro 

the United States and are not reserved to the people. 

(3) The powers reserved to the people are not delegated to 

the United States and are not reserved to the States. 

(4) The powers not delegated to the United States are 

either reserved to the States and not reserved to the 
people, or they are not reserved to the States and 
are reserved to the people. 

(5) The powers not reserved to the States are either not 

delegated to the United States and are reserved to 
the people, or they are delegated to the United 
States and not reserved to the people. 

(6) The powers not reserved to the people are either dele- 

gated to the United States and not reserved to the 
States, or they are not delegated to the United 
States and are reserved to the States. 

It will be seen that we have obtained from the conclusions 
which were given to us, twice as many propositions as we 
started with, and out of these six propositions thus obtained, 
we can get three sets of premises which will produce the 
original propositions; we can take the definitions of A and a, 
or B and b, or C and c. 

238. Of course in any given case it would be impossible for 
us to say which set of premises were the ones which the person 
who proposed the conclusions took for the purpose of obtaining 
those conclusions. We know that we have produced all the 
premises and that among the premises thus produced he must 
have taken two or more to obtain the conclusions originally 
given to us. 

239. Just as Deduction gives us a great many more con 
elusions than we need, so Induction gives us more premises 
than we need, and we have to make a selection, but the select ion 
which we make may not be the selection that some one else 
would make. It is enough, however, that we have produced all 
the possible premises. This example proves again, that there 
is no real difference between Deduction and Induction. In 



100 INDUCTION. [Chap. 8. 

either case we obtain a number of propositions which are con- 
sistent with the propositions which were given to us. 

240. A proposition may be either a premise or a conclusion. 
If we take it as a premise, then we call the consistent proposi- 
tions obtained from it, conclusions; if we take it as a conclu- 
sion, then we call the consistent propositions obtained from it, 
premises. 

241. In reasoning, Jill that we can do is to obtain other 
propositions. Or, in other woFds, given any proposition, rea- 
son enables us to discover all the different ways in which we 
can say the same thing. 

242. We cannot make any progress, we cannot discover any 
new facts; we can turn a proposition over and over again, 
exhibit it in a great many new lights, and from each new posi- 
tion get a new proposition, but all the propositions which we 
obtain, describe either the same idea or its opposite, and it fol- 
lows that, in all cases, the descriptions of an idea and of its 
opposite are equivalent, that is, from the one we can always 
obtain the other. 

243. If we start with a proposition like "Salt is chloride of 
sodium," we know that both these terms describe the same 
thought and are equivalent. We know that from the proposi- 
tion "Salt is chloride of sodium," we can obtain the equivalent 
proposition that not-salt is not-chloride of sodium, and the prop- 
osition not-salt is not-chloride of sodium will produce the prop- 
osition that salt is chloride of sodium, therefore, these proposi- 
tions are equivalent. 

244. But the one proposition describes an idea and the other 
describes its opposite and it follows that a proposition which 
describes an idea is equivalent to a proposition which describes 
the opposite, and vice versa. , 

245. When we say that the problem of logic is the problem 
of discovering all the hidden and latent meaning of proposi- 
tions, we really mean that it is the problem of discovering in 
how many different ways we can state the proposition. 

246. The problem of logic is, given an idea and its opposite 
and a description of either of them, either by name, title, desig- 



§§ 247, 248.] UNIVERSE OF DISCOURSE. 101 

nation or description, to find all the consistent descriptions 
of the idea and of its opposite that it is possible to discover. 

247. Our Universe of Discourse is always limited to one 
idea and its opposite, though of course we may change from one 
Universe of Discourse to another as often as we please, but a 
new Universe of Discourse implies a new idea, or field of 
thought. 

248. Let us return to our example. The conclusions given 
us were, 

(1) The powers delegated to the United States by the Con- 

stitution are not reserved to the States and are not 
reserved to the people. 

(2) The powers reserved to the States are not reserved to 

the people. 

(3) The powers not reserved to the States and not reserved 

to the people are delegated to the United States. 

We stated these propositions symbolically and analyzed them 
into their elements, and by our diagram showed every possible 
combination which could be made with the terms of these prop- 
ositions. We then eliminated all the inconsistent propositions 
and from the consistent propositions which remained we 
selected six. These six contained definitions of all the terms 
in the original proposition. We then said that the defini- 
tion of any one of these terms and its opposite would pro- 
duce the original conclusions. Let us prove this proposition. 

We will take these two propositions: 

(1) The powers delegated to the United States are not 

reserved to the States and are not reserved to the 
people. 

(2) The powers not delegated to the United States are 

either reserved to the States and not reserved to I he 
people, or they are not reserved to the States and are 
reserved to the people. 
Let A = powers delegated to the United States, 
b = the powers not-reserved to the States, 
c = the powers not-reserved to the people, 
a = the powers not-delegated to the United States, 



102 



INDUCTION. 



[ Chap. 8. 



B = the powers reserved to the States, 

C = the powers reserved to the people. 
In the two definitions taken, we have a definition of the 
powers delegated to the United States and a definition of the 
opposite idea, — the powers not-delegated to the United States. 
The definitions can be stated as, 

(1) A = Abe 

(2) a = aBc | abC 

Now, if A is Abe, then every combination of A which 
contains B or C or both, is inconsistent and must be eliminated ; 
we therefore eliminate the combinations ABC, AbC, ABc. 

Make an ABC diagram and eliminate those combinations by 
making a figure 1 in those sections, thus: 

AB Ab aB ab 



1 


1 


- 




1 






~ 



Fig. 45. 



Again, if a is aBc or abC, then the combinations aBC and abc 
are inconsistent and we eliminate them by making a figure 2 in 
those sections. 

The combinations Abc, aBc and abC automatically remain, 
and from them we can get the following definitions: 

(1) A = Abc 

(2) B = Bac 

(3) be = Abc 
Which may be translated, 

(1) The powers delegated to the United States are not 
reserved to the States and are not reserved to the 
people. 



§249] 



AN EXAMPLE. 



103 



(2) The powers reserved to the States are not delegated to 

the United States and are not reserved to the 
people. 

(3) The powers not reserved to the States and not reserved 

to the people are delegated to the United States. 

By omitting from the second definition the words "are not 
delegated to the United States" our premises have produced t he 
original conclusions. Of course when a definition contains 
more than is necessary for us to use, we may omit in our read- 
ing the part we do not need. 

249. Next, let us take the definitions of the powers reserved 
to the States, and the powers which are not reserved to the 
States. 

(1) The powers reserved to the States are not delegated to the 
United States and are not reserved to the people. 

It may be stated thus: 

(1) B = Bac 

(2) The powers not reserved to the States are either not dele- 
gated to the United States and are reserved to the people, or 
they are delegated to the United States and not reserved to the 
people. 

It may be sAted thus : 

(2) b = baC | bAc 
Make an ABC diagram as usual : 



AB 


Ab 


aB 


ab 


1 


o 


1 




1 






2 



Fig. 4(>. 

Now, if B is Bac, then the combinations of B which contain A 
or C or both, are inconsistent and must be eliminated. We 
therefore eliminate ABc and aBC and ABC, by making a figure 
1 in those sections. 



104 



INDUCTION. 



[ Chap. 8. 



Again, if b is baC | bAc, then the combinations AbC 
and abc are inconsistent, and we eliminate them by making a 
figure 2 in those sections. The combinations Abc, aBc, abC, 
automatically remain, and as they are identical with the defini- 
tions obtained from the definitions of "The powers delegated to 
the United States and not-delegated to the United States," they 
may be translated in the same manner that we translated them 
in the last preceding section. 

250. Now let us take the definitions of The powers reserved 
to the people, and of The powers not-reserved to the people. 

(1) The powers reserved to the people are not delegated to the 
United States, and are not reserved to the States. 

It may be stated thus: 

(1) C = Cab 

(2) The powers not-reserved to the people are either dele- 
gated to the United States and are not-reserved to the States, 
or, they are not delegated to the United States and are reserved 
to the States. 

It may be stated thus: 

(2) o = cAb | caB 
Make an ABC diagram, as usual: 

♦ 

AB Ab aB ab 



1 


1 


1 




2 






2 



Fig. 47. 



Now if C is Cab, then any combination of C with A or B, or 
both, is inconsistent. We therefore eliminate ABC, AbC, aBC 
by making a figure 1 in those sections. 

Again, if c is cAb or caB, then the combinations ABc and 
abc are inconsistent, and we eliminate them by making a fig- 
ure 2 in those sections. 



§ 251.J 



THE TENTATIVE METHOD. 



105 



As in the other cases, the combinations Abe, aBc, abC, 
remain, and from them we can get our original conclusions. 
Thus: 

(1) A = Abc 

(2) B = Bac 

(3) be = Abe 
And they may be translated: 

(1) The powers delegated to the United States by the Cou- 
stitution are not reserved to the States and are not reserved to 
the people. 

(2) The powers reserved to the States are not reserved to the 
people. 

(3) The powers not reserved to the States and not reserved to 
the people are delegated to the United States. 

Thus, we have proven that, given any conclusions, by obtain- 
ing from those conclusions the definitions of any term and its 
opposite, which are contained in the conclusions, and by using 
these definitions as premises, we can always obtain the original 
conclusions. 

251. Now let us try the tentative plan of obtaining pre- 
mises for the given conclusions. Let us take the same con- 
clusions as in the last section. The definition "The powers 
delegated to the United States are not reserved to the States 
and are not reserved to the people," may be stated thus: 
(1) A = Abe 



Make an ABC diagram: 



AB Ab aB ab 



1 

2 


1 


2 




1 
2 
3 






3 



Fig. 48. 



106 INDUCTION. [ Chap. 8. 

Now if A is Abc, then the combinations ABC, ABc and AbC 
are inconsistent and we eliminate them by making a figure 1 in 
those sections. 

(2) The definition "The powers reserved to the States are not- 
delegated to the United States and are not-reserved to the 
people/' may be stated thus: 

(2) B = Bac 

Now if B is Bac, then the combinations ABC, ABc, aBC, are 
inconsistent, and we eliminate them by making a figure 2 in 
these sections. 

Now, in order to obtain the given conclusions, we must get 
rid of the combination abc. Of course a definition of C would 
not enable us to get rid of c, because a definition of c would not 
be inconsistent witli a definition of C. We must, therefore, 
get a definition of c. 

(1) "The powers not reserved to the people are either dele- 
gated to tin' United States and not reserved to the States, or 
they are not delegated to tli*- United States and are reserved to 
the States."' 

It may be stated thus: 

(3) c = cAb | caB 

If c is cAb or call, then the combinations ABc and abc are 
inconsistent, and we eliminate them by making a figure 3 in 
those sections. 

Our original conclusions again remain. They are: 

(1) A = Abc 

(2) B = Bac 

(3) be = Abc 

Thus by the tentative process we have found that the defini- 
tions of the powers delegated to the United States, and of the 
powers reserved to the States, and of the powers not reserved 
to the people, have given us the original conclusions. 

252. Frof. Jevons, in his Principles of Science, p. 125, 
speaking of the problem of Induction, that is, of finding pre- 
mises for given conclusions, says: "The only modes of discov- 



§§ 253-256.] POSSIBLE COMBINATIONS. 107 

ery consist either in exhaustively trying a great number of 
supposed laws, a process which is exhaustive in more senses 
than one, or else in carefully contemplating the effects and 
endeavoring to remember cases in which like effects followed 
from unknown laws." 

253. Dr. Keynes, in his work on Formal Logic, shows that 
the Inductive problem may be solved in the way which I have 
given. 

251. Again, on p. 137 of Principles of Science, Jevons says: 
"Now, we may make selection from eight things in two hun- 
dred and fifty-six ways; so that we have no less than tw r o hun- 
dred and fifty-six different cases to treat, and the complete 
solution is at least fifty times as troublesome as with two 
terms." It may take more time to solve a problem involving 
three terms than it takes to solve a problem involving only two 
terms, but it is not much more troublesome, it merely takes a 
little more time. 

255. Again, he says on p. 141: "The above investigations 
are complete as regards the possible logical relations of two 
or three terms. But when we attempt to apply the same kind 
of method to the relations of four or more terms, the labor be- 
comes impracticably great. Four terms give sixteen combina- 
tions compatible with the laws of thought, and the number of 
possible selections of combinations is no less than G5,536. 
Some years of continuous labor would be required to ascertain 
the types of laws which may govern the combinations of only 
four things, and but a small part of such laws w T ould be exem- 
plified or capable of practical application in science. The 
purely logical inverse problem whereby we pass from combi- 
nations to their laws, is solved in the preceding pages, as far 
as it is likely to be for a long time to come, and it is almost 
impossible that it should ever be carried more than a single 
step further." 

256. The Inductive problems containing four or more terms 
are not much more troublesome to solve than those containing 
two or three terms. Instead of "some years," it can only 



108 INDUCTION. [ Chap. 8. 

take a few minute's time and labor to find the premises which 
will produce the conclusions containing four or more terms. 

257. Miss Jones in her Elements of Logic, p. 60, points out 
very clearly a fallacy in Inductive reasoning made by some of 

the old logicians. She says: "Snnday, Monday and 

Saturday are all (omnes) twenty-four hours in length; Sunday, 

Monday and Saturday are all (cnncti) the days of the 

week. 

Therefore, all (omnes) the days of the week are twenty-four 
hours in length. It may be remarked that this syllogism is 
incorrect in form, the minor terms being taken collectively in 
its premise, distributive^ in the conclusion. I do not remem- 
ber to have seen this inaccuracy noted. 

Mansel (MansePs Aldrich, 4th Ed., p. 221), Whately (Logic, 
0th Ed., p. 152), and Jevons (Elementary Lessons, 7th Ed., pp. 
214 and 215) among others, offer as instances of perfect or 
Aristotelian Induction, arguments exactly corresponding in 
form to the one I have given, without any remark upon their 
formal incorrectness." 

258. Induction, as it is treated by the old logic, does not 
seem to me to be logical at all. According to the old logic, we 
state the instances which have come under our observation, 
and from these instances we infer a general law. From "Some 
A is B," we infer all A is B, according to the old logic. But 
this is not inference; this is making an unwarranted jump. Of 
course it may be true that all A is B, but it is not a logical 
inference from some A is B. The author of the article on 
Logic in the Encyclopedia Britannica, says: "Induction 
makes clear only, and does not prove." He is speaking of the 
old logic. 

259. There are two other problems in reasoning which our 
system enables us to solve very easily. One is, given any prop- 
osition, how can we prove it to be true? The other is, given 
any proposition, how can we prove it to be false? Let us take 
this example: Suppose we are given the proposition, "Iron is 
me^al-element." 



§ 259.] 



TRUTH AND FALSITY. 



109 



Let A = iron, 
B == metal, 
C = element. 
As the predicate is undistributed, that is, it does not mean 
that all metallic elements are iron, we can state it thus: 

A = ABC. 
Make an ABC diagram : 

AB Ab aB ab 





1 






1 


1 







Fig. 49. 



Now, if ABC is true, this would cause us to eliminate the 
other three combinations of A, viz.: ABc, AbC, Abe. The^p 
may be translated, 

(1) Iron is metal, not-element. 

(2) Iron is not-metal element. 

(3) Iron is not-metal and not-element. 

Now, we can prove conclusively that A is ABC, by proving 
that ABc and AbC and Abe are false. There are only four 
combinations of A; if one is true the other three must be false; 
proving that the three are false, proves that the other one is 
true. 

In order to prove that ABC is false, it is only necessary for 
us to prove that either ABc or AbC or Abe is true. In this 
case the truth of any one of the propositions which ABC would 
cause us to eliminate, proves the falsity of ABC. Suppose I his 
proposition is given us: A is either ABC or Abe. This would 
cause us to eliminate the other two combinations of A. viz.: 
AbC and ABc. Now, we can prove that A is ABC, or Abe by 
proving that AbC and ABc are false. Again, W€ can prove the 



110 



INDUCTION. 



[ Chap. 8. 



falsity of the statement that A is ABC or Abe by proving the 
truth of either of the statements that A is AbC or that A is 
ABc. 

260. This leads us to consider what propositions are equiv 
alent to each other. Propositions which are equivalent to each 
other will cause us to strike out and save exactly the same com- 
binations from our diagram. 

261. Prof. Jevons in his Principles of Science, p. 11G, says: 
"In the following list each proposition or group of propositions 
is exactly equivalent in meaning to the corresponding one in 
the other column, and the truth of this statement may be tested 
by working out the combinations of the alphabet." 





A — Ab 




B = aB 




A = b 




a = B 




A = BC 




a = b | c 




A = AB | AC 




b — ab | AbC 


A 


| B = C | D 




ab — > cd 


A 


j c — B | d 




aC — bD 




A = ABc | AbC 


i 


A — AB | AC 

AB — ABc 




A — B I 
B = C J 


I 


A = B 

A — A 




A = AB ) 
B — BC ) 


i 


A — AC 

B = AC 1 aBC 



The first two examples are correct. Let us try the third. 
:>Jake an ABC diagram: 



AB Ab aB ab 





1 


8 




1 


1 







Fie. 50. 



§ 261.] 



EQUIVALENT PROPOSITIONS. 



Ill 



Now, If A equals BC; AbC, Abe and ABc are inconsistent 
and we eliminate them by making a figure 1 in those sections. 

And if BC equals A, reading the proposition backward, then 
aBC is inconsistent and we eliminate it by making a figure 2 
in mat section. Now make an ABC diagram: 

AB Ab aB ab 







1 










1 



Fig. 51. 

The proposition a equals b or c, may have two different 
meanings, depending upon whether we consider "or" as mean- 
ing one or the other and not both, or as meaning one or the 
other or both. Taking the first meaning it should be stated, 

a = bC | Be 
Taking the second meaning it should be stated, 

a = bC | Be | be 
If the first meaning is correct, the combinations aBC and abe 
are inconsistent and we eliminate them by making a figure 1 
in those sections. Make another ABC diagram: 

AB Ab aB ab 







1 













Fig. 52. 



If the second meaning is correct, it will cause us to eliminate 
aBC and we eliminate it bv making a figure 1 in that section. 



112 



INDUCTION. 



[Chap. 8. 



The appea ranee of the three diagrams shows that Jevons' 
statement in this case is incorrect. We shall now pass on to 
the last example given by Jevons. 

262. The last example reads, 

A — AB, ) , 3 1 a ( A = AC, 

^ ^^ V the second column reads ■{ ^ , '„ 

B = BC j j B = A | aBC, 

Jf A is AB, AbC and Abe are inconsistent and we eliminate 
them by making a figure 1 in those sections: 



AB Ab aB ab 





1 






2 


1 


2 





Fig. 53. 



Again, if B is BC, the combinations ABc and aBc are incon- 
sistent and we eliminate them by making a figure 2 in those 
sections. Now make another ABC diagram: 



AB Ab aB ab 











1 


1 


o 





AB Ab aB ab 



Fig. 54. 



Now. if A is AC, then the combinations ABc and Abe are 
inconsistent and we eliminate them by making a figure 1 in 
those sections. 

Again, if B is A or aBC, then the combination aBc is incon- 



§ 263.] 



EQUIVALENT PROPOSITIONS. 



113 



sistent and we eliminate it by making a figure 2 in that sec- 
tion. The appearance of the two diagrams again shows a mis- 
take on Jevons' part. 

263. Let us take these two examples which were incorrectly 
solved by Prof. Jevons, and show the correct solution by our 
system. Make an ABC diagram: 

AB Ab aB ab 





1 


2 




1 


1 







Fig. 55. 



Now, if A is BC, we eliminate the combinations ABc, AbO, 
Abe, as inconsistent, by making a figure 1 in those sections. 

And if BG is A, then aBG is inconsistent and we eliminate it 
by making a figure 2 in that section. 

Now, as we showed before in treating of the Inductive prob- 
lem, if from the conclusions which remain as consistent, we get 
the definition of any term and its opposite, these definitions 
will cause us to strike out the same combinations which, when 
struck out before, gave us the conclusions. 

Now, in this case, the conclusions which remain are ABC, 
aBc, abC and abc. The definitions of B and b are, 

(2) b = abC | abc 
(1) B = ABC 

The definitions of C and c are, 

(3) C = ABC 

(4) C = aBc | 
The definitions of A and a are, 

(5) A=.-ABC 

(6) a = aBc | 
8 



| aBc 

| abC 
abc 



abC I abo 



114 



INDUCTION. 



[ Chap. 8. 



Either pair of definitions will cause us to strike out exactly 
the same combinations that the proposition A is BC will cause 
us to strike out, therefore, either pair of definitions is the exact 
equivalent of the proposition A is BC. By the tentative 
piocess, already explained, we could get other equivalents for 
the given proposition but none of them can possibly resemble 
the equivalent which Jevons gives. We will work out the 
pans of definitions given above, on the diagram: 

AB Ab aB ab 





2 


1 




1 


9 







Fig. 56. 



If 1* is ABC or aBc. then the combinations aBC and ABc are 
inconsistent and we eliminate them by making a figure 1 in 
those sections. 

If b is abC or abc, then the combinations AbC and Abe are 
inconsistent and we eliminate them by making a figure 2 in 
those sections. 

And now we have struck out exactly the combinations which 
the proposition A is BC caused us to eliminate. 

Again, if C is ABC or abC, then the combinations AbC and 
aBC are inconsistent and we eliminate them by making a figure 
1 in those sections. 

AB Ab aB ab 





1 


1 




2 


2 







Fig. 57. 



§264.] 



EQUIVALENT PROPOSITIONS. 



115 



If c is aBc or abc, then the combinations ABc, and Abe are 
inconsistent and we eliminate them by making a figure 2 in 
those sections. And again, we have struck out the same sec- 
tioi;S which the original proposition A is BC caused us to 
eliminate. 

Now make another ABC diagram : 

AB Ab aB ab 





1 


2 




1 


1 







Fig. 58. 

If A is ABC, then the combinations ABc, AbC and Abe are 
inconsistent and we eliminate them by making a figure 1 iu 
those sections. 

If r. is aBc, or abC, or abc, then the combination aBC is incon- 
sistent and we eliminate it by making a figure 2 in that section. 

Again we have struck out the same sections which the orig- 
inal proposition, A is BC, caused us to eliminate. This proves 
that either pair of definitions given is equivalent to the original 
proposition. 

2G-1. We will now take Jevons' last example. Make an 
A BC diagram : 

AB Ab aB ab 



1 

1 
2 2 



Fig. 59. 



116 



INDUCTION. 



[ Chap. 8. 



If A is AB, then the combinations AbC, Abe are inconsistent 
and we eliminate them by making a figure 1 in those sections. 

If 13 is BC, then the combinations A Be, aBc are inconsistent 
and we eliminate them by making a figure 2 in those sections. 

The propositions which remain in the diagram are ABC, 
aBC, abC, abc. If from these propositions which remain in 
the diagram, we get the definitions of any term and its oppo- 
site, those definitions will cause us to strike out exactly the 
same combinations which the original proposition caused us 
to strike out. They will, therefore, be the true equivalents of 
the original propositions. The definitions which we can obtain 
are: 

(1) A = ABC 

(2) a = aBC | abC | abc 

If A is ABC, then the combinations ABc, AbC, Abc are incon- 
sistent and we eliminate them by making a figure 1 in those 
sections: 

AB Ab aB ab 





1 






1 


1 


2 





Fig. 60. 



If a is aBC, or abC, or abc, then the combination aBc is 
inconsistent and we eliminate it by making a figure 2 in that 
section. 

Thus, the definitions of A and a have caused us to strike out 
the same combinations which the propositions A is AB and B 
is BC caused us to eliminate. 

Kow make another ABC diagram: 



§ 264.] 



EQUIVALENT PROPOSITIONS. 
AB Ab aB ab 



117 





2 






1 


2 


1 





Fig. 61. 

The definitions of B and b are, 

(3) B = ABC | aBO 

(4) b = abO | abc 

If B is ABC, or aBC, then the combinations ABc, aBc, are 
inconsistent and we eliminate them by making a figure 1 in 
those sections. 

If b is abC, or abc, then the combinations AbC, Abc are 
inconsistent and we eliminate them by making a figure 2 in 
those sections. 

Again we have struck out the same combinations which the 
original propositions A is AB and B is BC caused us to strike 
out. 

The definitions of C and c are, 

(5) C = CAB | CaB | Cab 

(6) c = abc 
Make another ABC diagram: 



AB 


Ab 


aB 


ab 




1 






2 


2 


2 





Fig. G2. 
If C is CAB or CaB or Cab, then the combination AbC is 



118 INDUCTION. [ Chap. 8. 

inconsistent and we eliminate it by making a figure 1 in that 
section. 

-Again, if c is abc, then the combinations ABc, Abe, aBc, are 
inconsistent and we eliminate them by making a figure 2 in 
those sections. 

Anc again we have struck out the same combinations which 
the original propositions caused us to strike out. This proves 
that either pair of definitions is the logical equivalent of the 
original propositions given to us, viz: A is AB and B is BC. 

EXERCISES. 

265. What conclusions can be drawn from the following 
premises? 

(1) A = AB 
C = Ca 

(2) a = cab 
C = Ca 

(3) C = Cb 
c = caB 

(4) C = CB 
c = abc 

(5) C = CAB 
c = ca 

(6) B = AC 
a = be 

(7) What premises will produce the conclusions 
ABC 
Abc 
aBc 
abC 



CHAPTER IX. 

A CHAIN OF REASONING. 

266. Thus far we have acted on the hypothesis that the prop- 
ositions which were given to us were true, but it is possible 
that a premise may be false. 

if a proposition is true, then all the propositions which are 
inferences from it are true, and all the propositions which are 
inconsistent with it are false. 

267. If a proposition is false, then all the propositions which 
are inferences from it are false, and at least one of the propo- 
sitions, which are inconsistent with it, is true. 

268. We can thus reason from the truth of a proposition to 
the truth or falsity of other propositions, and from the falsity 
of a proposition we can reason to the falsity of other proposi- 
tions and to the truth of at least one proposition. Let us 
illustrate this by a concrete example. 

269. Let us take this example: The powers which are dele- 
gated to the United States are not reserved to the States and 
are not reserved to the people. 

Let A = the powers delegated to the United States, 
b = the powers not reserved to the States, 
c = the powers not reserved to the people. 
We take it that the predicate is distributed, that is, that jthe 
powers not reserved to the States and not reserved to the 
people are delegated to the United States, so that we state 
the proposition both ways, 

(1) A = be 

(2) be = A 

270. It is proper to observe that in stating propositions we 
drop the conjunctions, prepositions and other connecting 
words, and when we come to read our symbolic propositions 
we supply whatever conjunctions, prepositions and other con- 



120 



A CHAIN OF REASONING. 



[Chap. 9. 



necting words may be necessary to make readable English. 
Our symbols represent only logical elements, and it sometimes 
takes considerable ingenuity to translate our symbols into 
good English. 

Now, if the proposition A is bo is true, then the other 
combinations containing A are false, viz.: ABC, ABc, AbC, 
and if the proposition be is A is true, then the combination abc 
is false. 

Make an ABC diagram: 



AB 


Ab 


aB 


ab 




1 


1 






C 


1 






2 






Fig. 63. 

This diagram will assist the reader to follow this demonstra- 
tion. We can translate the letters in the preceding paragraph 
thus: 

If the proposition, 

(1) The powers which are delegated to the United States are 
not reserved to the States and are not reserved to the people, 
(A = be) be true, then the propositions, 

(2) The powers which are delegated to the United States are 
reserved to the States and are reserved to the people (A = BC) 
and the proposition, 

(3) The powers which are delegated to the United States are 
reserved to the States and are not reserved to the people (A = 
Be), and the proposition, 

(4) The powers which are delegated to the United States are 
not reserved to the States and are reserved to the people (A = 
bC), are false; and if the proposition, 

(5) The powers which are not reserved to the States and 
which are not reserved to the people are delegated to the 
United States (be = A), be true, then the proposition, 



§270.] AN EXAMPLE. 12 i 

(6) The powers which are not reserved to the States and 
which are not reserved to the people, are not delegated to the 
United States (be = a), is false. 

It will be necessary for us to translate the symbolic propo- 
sitions which follow. They are all to be interpreted similarly 
to the preceding, and it will save us time and labor to work 
with symbols only, in the first instance. 

(1) If the proposition A is ABC is false, then one of the 
combinations ABc, AbC, Abe is true. 

(2) If the proposition A is ABc is true, then the combinations 
ABC, AbC, Abe are false. 

(3) If the proposition A is AbC is false, then one of the com- 
binations ABC, ABc, Abe is true. 

(4) If the proposition A is Abe is false, then one of the com- 
binations ABC, ABc, AbC is true. 

(5) If the proposition A is ABC is true, then the combina- 
tions ABc, AbC, Abe are false. 

(6) If the proposition A is ABc is false, then one of the com- 
binations ABC, AbC, Abe is true. 

(7) If the proposition A is AbC is true, then the combina- 
tions ABC, ABc, Abe are false. The translations are as fol- 
lows: 

If the proposition, 

(1) The powers delegated to the United States are not 
reserved to the States and are not reserved to the people (A — 
be), is false, then one of the combinations, 

(a) The powers which are delegated to the United States are 
reserved to the States and not to the people (A == Be), 

or, 

(b) Are not reserved to the States and are reserved to the 
people (A = bC), 

or, 

(c) Are not reserved to the States and not reserved to the 
people (A = be), 

is true. 

If the proposition, 



122 A CHAIN OF REASONING. [Chap. 9. 

(2) The powers which are delegated to the United States are 
reserved to the States and not to the people (A = Be), 

is true, then all the combinations, 

(a) The powers which are delegated to the United States are 
reserved to the States and to the people (A = BC), 

and, 

(b) Are not reserved to the States but are reserved to the 
people (A = bC), 

and, 

(c) Are not reserved to the States and not reserved to the 
people (A = be), 

are false. 

If the proposition, 

(3) The powers which are delegated to the United States are 
not reserved to the States, but are reserved to the people (A 
= bC), 

is false, then one of the combinations, 

(a) The powers which are delegated to the United States are 
reserved to the States and to the people (A = BC), 

or, 

(b) Are reserved to the States and not to the people (A =Bc), 
or, 

(c) Are not reserved to the States and not reserved to the 
people (A = be), 

is true. 

If the proposition, 

(4) The powers which are delegated to the United States are 
not reserved to the States and not reserved to the people (A = 
be), 

is false, then one of the combinations, 

(a) The powers which are delegated to the United States are 
reserved to the States and to the people (A = BC), 

or, 

(b) Are reserved to the States and not to the people (A = 
Be), 

or, 



§ 270.] AN EXAMPLE. 12 3 

(c) Are not reserved to the States but to the people (A = bC), 
is true. 

If the proposition, 

(5) The powers which are delegated to the United States are 
reserved to the States and to the people (A = BC), 

is true, then all the combinations, 

(a) The powers which are delegated to the United States are 
reserved to the States and not to the people (A = Be), 

and, 

(b) Are not reserved to the people but are reserved to the 
States (A == bC), 

(c) Are not reserved to the States and not to the people (A = 
be), 

are false. 

If the proposition, 

(6) The powers which are delegated to the United States are 
reserved to the States and not to the people (A = Be), 

is false, then one of the combinations, 

(a) The powers which are delegated to the United States are 
reserved to the States and to the people (A = BC), 

or, 

(b) Are not reserved to the States but are reserved to the 
people (A = bC), 

or, 

(c) Are not reserved to the States and not to the people (A = 
be), 

is true. 

If the proposition, 

(7) The powers which are delegated to the United States 
are not reserved to the States but are reserved to the people 
(A = bC), 

is true, then all the combinations, 

(a) The powers which are delegated to the United States are 
reserved to the States and to the people (A = BC), 
and, 



124 A CHAIN OF REASONING. [Chap. 9. 

(b) Are reserved to the States and not to the people (A = 
Be), 

and, 

(c) Are not reserved to the States and not to the people (A = 
be), 

are false. 

We have now considered the propositions in which the sub- 
ject is, 

(A) "The powers which are delegated to the United States," 
as to their truth or falsity, and we have found that there were 
eight different combinations. So that these propositions which 
have the proposition, 

"The powers which are delegated to the United States," 
for their subject, have made quite a chain of reasoning. There 
are eight links in the chain. But we could go on and consider 
next, propositions which have, 

(a) "The powers which are not delegated to the United 
States," 

for their subject. 

Then we can take up in order the propositions which have 
for their subjects respectively, 

(B) "The powers which are reserved to the States," 
and, 

(b) "The powers which are not reserved to the States," 
and then the propositions which have for their subject, 

(C) "The powers which are reserved to the people," 
and, 

(c) "The powers which are not reserved to the people," 

and then the propositions which have for their subjects, 
respectively, 

(AB) "The powers which are delegated to the United States 
and are reserved to the States." 

(Ab) "The powers which are delegated to the United States 
and are not reserved to the States," 

(aB) "The powers which are not delegated to the United 
States and are reserved to the States." 



§§ 271-273.] AN EXAMPLE. U ' D 

(ab) "The powers which are not delegated to the United 
States and not reserved to the States," 

and then the propositions which have for their subjects, 
respectively, 

(BC) "The powers which are reserved to the States and to 
the people." 

(Be) "The powers which are reserved to the States but not 
to the people." 

(bC) "The powers which are not reserved to the States and 
are reserved to the people." 

(be) "The powers which are not reserved to the States and 
not to the people." 

and then the propositions which have for their subjects, respec- 
tively, 

(AC) "The powers which are delegated to the United States 
and are reserved to the people." 

(Ac) "The powers which are delegated to the United States 
but not reserved to the people." 

(aC) "The powers which are not delegated to the United 
States but are reserved to the people," 

(ac) "The powers which are not delegated to the United 
States and not reserved to the people." 

271. Now, as there are eighteen different subjects, which 
we may call eighteen chains of reasoning, and as there are 
eight links in each chain, when the chains are joined we have 
a chain with 144 links. 

272. We suggest to the reader that if he wishes to obtain 
an exhaustive view of all questions which are contained in a 
single proposition involving three terms, that he pursue the 
same course with the other subjects that we have pursued with 
the subject, "The powers which are delegated to the United 
States." 

273. It is clear that we can also reason from the truth of a 
proposition to the truth of other propositions, and from the 
falsity of a proposition to the falsity of other propositions. 
With the case in hand we can obtain just as many subjects 



12G A CHAIN OF REASONING. [ Chap. 0. 

and just as many propositions, in reasoning from truth to truth 
and falsity to falsity, as we did in reasoning from truth to falsity 
and from falsity to truth. This will giye us another chain with 
144 links in it, which added to the other chain would make a 
chain of 288 links. As there are four combinations in each 
link, the total number of combinations would be 1152. These 
combinations, consisting each of three terms, can be read in 
various ways, according to the Law of Permutations. I haye 
not figured it out, but I am under the impression that our sys 
tern would give us oyer 25,000 different readings of the given 
example. It shows the immense variety which can be obtained 
from a very few terms. 

274. In reasoning from the truth of a proposition to the 
truth of other propositions, of course all the propositions which 
are inferences from the given proposition, will be true, and in 
reasoning from the falsity of a proposition to the falsity of 
other propositions, all the propositions which are inferences 
from the given proposition, will be false. 



CHAPTER X. 

TERMS. 

275. When the subject or predicate of a proposition can be 
represented by a single letter, we call the subject or predicate 
a single term, but when it is necessary to represent either of 
them by two or more letters, viz.: AB, BC, etc., we call the 
combination a complex term. 

276. Single terms can be combined conjunctively or disjunc- 
tively. In the proposition A is BC, the subject A is a siugle 
term, the predicate BC is aconjunctive term and it means B and 
C. In the proposition A is B or C, the predicate is a disjunc- 
tive term and it means A is B without C, or C without B and its 
symbolic expression is, A is Be or Cb. 

277. The use of negative letters is necessary to the full 
logical expression of a disjunctive term, but in reading a dis- 
junctive term it is usual to omit the negative terms, when the 
use of positive terms will express the meaning. Similarly, 
when in order to state a proposition logically, it is necessary 
to repeat the letter which represents the subject in the predi- 
cate, it is not necessary, in reading, to repeat the letter which 
represents the subject. 

278. It has been suggested that the letters in the conjunc- 
tive term, e. g., AB, be called determinants. A and B are the 
determinants in the conjunctive term AB. 

279. It has also been proposed that the single letters in a 
disjunctive term be called the alternants. Thus, in the propo- 
sition A is B or C, B and C are the alternants, but this is Dot 
strictly correct. The proper expression of A is B or C is, 
A = Be | bC, so that strictly the alternants are Be and bC. 

280. In order to fully express the logical meaning of an 
alternant, we must use both positive and negative deter- 
minants. 



128 



TERMS. 



[Chap. 10. 



281. The order of stating and reading the determinants 
in a conjunctive or disjunctive term is a matter of indifference. 
We can state and read them forward or backward or in any 
order we please. 

282. In this system we understand by the expression A is 
B or 0, that A cannot be both B and C or neither. 

283. When we say A or B is C, we mean, in this system, 
that A without B, or B without A, is C, and its symbolic expres- 
sion is, Ab or aB is C. It excludes the idea that both A and I> 
or neither, can be C. 

284. The question has been asked what is the contradic- 
tory of AB? 

Make an AB diagram. 



A 


a 






1 


B 


1 


1 


b 



Fig. 64. 

285. I do not think that terms have contradictories; terms 
have opposites, but propositions have contradictories; the ques- 
tion should be what is the opposite of AB. 

286. When we have eliminated the inconsistent combina- 
tions we can read them, "No combinations are," naming them. 
Or we can substitute for the expression, "No combinations 
are," the shorter phrase, "Nothing is." 

287. We can read the uneliminated combinations, "All the 
combinations are," or "Every combination is," naming them. 
For these expressions we can substitute the shorter phrase, 
everything is, understanding that "everything" stands for 
every combination. 

288. In. order to solve this problem we suppose AB to be 
the only combination uneliminated in the Reasoning Frame. 



§§ 289, 290.] 



OPPOSITE TERMS. 



129 



Now, if everything is AB, then the combinations Ab, aB, ab 
are inconsistent and we eliminate them by making a figure 1 
in those sections. 

We can now read in the Reasoning Frame, 

(1) Everything is AB 

(2) Nothing is Ab or aB or ab, which can be reduced to, 
Nothing is Ab or a. 

Hence the opposite of AB is Ab | a. 

Everything is AB is an impossible proposition because it 
causes the elimination of a and b. 

289. What is the opposite of a or b? 
Make an AB diagram. 



A 


a 




1 




B 




1 


b 



Fig. 65. 

Now, if everything is aB or Ab, then the combinations AB, 
ab are inconsistent and we eliminate them by making a figure 
1 in those sections. 

We can now read in the Reasoning Frame, 

(1) Everything is a or b 

(2) Nothing is AB or ab. Hence the opposite of a or b is A B 
or ab. 

290. What is the opposite of A or B? The expression of A 
or B means, A without B, or B without A, and it can be 
expressed Ab or aB. 

Make an AB diagram. 



9 



130 



TERMS. 



[Chap. 10. 



A 


a 




1 




B 




1 


b 



Fig. 66. 

Now, if everything is Ab or aB, then the combinations AB, 
ab are inconsistent and we eliminate them by making a figure 
1 in those sections. 

We can now read in the Reasoning Frame, 

(1) Everything is A or B 

(2) Nothing is AB or ab. Hence the opposite of A or B is 
AB or ab. 

291. What is the opposite of ab? 
Make an AB diagram. 



A 


a 




1 


1 


B 


1 




b 



Fig. 67. 

Now, if everything is ab, then the combinations AB, Ab, aB 
are inconsistent and we eliminate them by making a figure 1 
in those sections. 

We can now read in the Reasoning Frame, 

(1) Everything is ab. This is an impossible proposition 
because A and B are eliminated. 

(2) Nothing is AB or Ab or aB. Hence the opposite of ab 
is AB or aB or Ab. 

292. What is the opposite of A or BO? 
Make an ABC diagram. 



§§ 293, 294.] 



OPPOSITE TERMS. 



131 



AB 


Ab 


aB 


ab 




1 






1 


C 






1 


1 


c 



Fig. 68. 

Now, if everything is A or BC, then the combinations ABC, 
aBc, abC, abc are inconsistent and we eliminate them by mak- 
ing a figure 1 in those sections. 

We can now read in the Reasoning Frame, 

(1) Everything is A or BC. 

(2) Nothing is ABC or aBc or abC or abc 

Hence the opposite of A or BC is ABC or aBc or abC or abc. 
293. What is the opposite of ab or ac? 
Make an ABC diagram. 



AB 


Ab 


aB 


aB 




1 


1 


1 




C 


1 


1 




1 


c 



Fig 69. 

Now, if everything is ab or ac, then the combinations con- 
taining AB, Ab, aBC, abc are inconsistent and we eliminate 
them by making a figure 1 in those sections. 

We can now read in the Reasoning Frame, 

(1) Everything is ab or ac 

(2) Nothing is ABC or ABc or AbC or Abc or aBC or abc 
Hence the opposite of ab or ac is aBC or abc or ABC or ABc 

or AbC or Abc. 
294. What is the opposite of ABC or ABD? 
Make an ABCD diagram. 



132 



TERMS. 



[ Chap. 10. 



AB 


Ab 


aB 


ab 




1 


1 


1 


1 


CI) 




1 


1 


1 


Cd 




1 


1 


1 


cD 


1 


1 


1 


1 


cd 



Fig. 7i). 

Now, if everything is ABC or ABD, then the combinations 
containing ABCD, ABcd, Ab, aB, ab are inconsistent and we 
eliminate them by making a figure 1 in those sections. 

We can now read in the Reasoning Frame, 

(1) Everything is ABC or ABD 

(2) Nothing is ABCD or ABcd or Ab or a. Hence the oppo- 
site of ABC or ABD, is ABCD or ABcd or Ab or a. 

295. What is the opposite of a or b or cd? 
Make an ABCD diagram. 



AB 


Ab 


aB 


ab 




1 






1 


CD 


1 






1 


Cd 


1 






1 


cD 




1 


1 


1 


cd 



Fig. 71. 

Now, if everything is a or b or cd, then the combinations con- 
taining ABC, ABcD, Abed, aBcd, ab are inconsistent and we 
eliminate them by making a figure 1 in those sections. 

We can now read in the Reasoning Frame, 

(1) Everything is a or b or cd 

(2) Nothing is ABC or ABcD or Abed or aBcd or ab. 
Hence the opposite of a or b or cd is ABC or ABcD or Abed or 
aBcd or ab. 



§§ 296, 297.] 



LOGICAL QONTRARIES. 



133 



296. Complex terms may be different without being oppo- 
sites. They are different when one contains a positive term 
and the other contains the negative of that positive term, 
thus: ABC and ABc are different without being opposites. 

297. Terms which have the determinant in one replaced by 
the opposite determinant in the other, are called logical contra- 
ries, thus AbC, aBc are called contraries. 

When we take a pair of contraries, e. g., AbC, aBc and get 
a definition of each letter stated in the remaining two letters, 
we have this interesting result, that the triplet of definitions 
furnished by one is equivalent to the triplet of definitions fur- 
nished by the other. Thus, from AbC we can get these defini- 
tions, 

A = AbC 

b = bAC 

C = CAb 
Make an ABC diagram. 



AB 


Ab 


aB 


ab 




1 
3 




3 


2 
3 


C 


1 


1 
2 




2 


c 



Fig. 72. 

Now, if A = AbC, then the combinations ABc, ABC, Abe 
are inconsistent and we eliminate them by making a figure 1 in 
those sections. 

Again, if b = bAC, then the combinations Abe, abC, abo 
are inconsistent and we eliminate them by making a figure 2 in 
those sections. 

Again, if C = CAb, then the combinations ABC, aBC, abC 
are inconsistent and we eliminate them by making a figure 3 in 
those sections. 



134 



TERMS. 



[ Chap. 10. 



298. From aBo we can get these definitions: 

a = aBc 
B = Bac 
c = caB 

Make an ABC diagram. 



AB 


Ab 


aR 


ab 




2 




•> 

1 


1 


C 


3 


3 




1 
3 


c 



Fig. 73. 



Now, if a = aBc, then the combinations abC, abc, aBC are 
inconsistent and we eliminate them by making a figure 1 in 
those sections. 

Again, if B = Bac, then the combinations ABC, ABc, aBO 
are inconsistent and we eliminate them by making a figure 2 in 
those sections. 

Again, if c = caB, then the combinations ABc, Abc, abc are 
inconsistent and we eliminate them by making a figure 3 in 
those sections. 

An examination of the two Frames now shows the equival- 
ence of the given triplets of definitions. 

299. Prof. Keynes in speaking of the obversion of complex 
propositions on p. 403, says, "The obverse, therefore, of all X 
is AB or ab, is no X is Ab or aB. This agrees with the conclu- 
sion which we have arrived at. 

300. The question is, What is an equivalent expression for 
ABor AC? 

Make an ABC Diagram. 



§ 301.] 



EQUIVALENT TERMS. 



135 



AB 


Ab 


aB 


ab 




1 




1 


] 


C 




1 


1 


1 


c 



Fig. 74. 

Now, if everything = AB | AC, then the combinations con- 
taining ABC, Abe, a, are inconsistent and we eliminate them 
by making a figure 1 in those sections. 

We can now read in the Reasoning Frame, 

(1) Everything = AB | AC 

(2) Everything = Ab | Ac 

(3) Nothing = ABC | Abe | a 

(4) C = Ab 
(5) c = AB 

Hence (2) and (3) are equivalents for (1). 

301. What is an equivalent expression for A or BC? 
Make an ABC Diagram. 



AB 


Ab 


aB 


ab 




1 






1 


C 






1 


1 


c 



Fig. 75. 



Now, if everything = A | BC, then the combinations contain- 
ing ABC, aBc, ab are inconsistent and we eliminate them by 
making a figure 1 in those sections. 

We can now read in the Reasoning Frame. 

(1) Everything = A | BC 

(2) Everything = Ab | ABc, | aBC 



136 



TERMS. 



[ Chap. 10. 



(3) Everything = Ac | AbC | aBC 

(4) Nothing = ABC | aBc | ab 
(C) Nothing = ac \ ABC | abC 

The result proves that (2), (3), (4) and (5) are equivalent ex- 
pressions for (1). 

302. What is an equivalent expression for A or aB? 
Make an AB diagram. 



A 


a 








B 






b 



Fig. 76. 

Now, if everything = A | aB, then the combination ab is 
inconsistent and we eliminate it by making a figure 1 in that 
section. 

We can now read in the Reasoning Frame, 

(1) Everything = A | aB 

(2) Everything = B | Ab 

(3) Nothing = ab 

The result proves that (2) and (3) are equivalent expressions 
for (1). 

303. What is an equivalent expression for AC or AD or BD? 
Make an ABCD diagram. 



AB 


Ab 


aB 


ab 




1 


1 


1 


1 


CD 


1 




1 


1 


Cd 


1 






1 


cD 


1 


1 


1 


1 


cd 



Fig. 77. 



§§ 304-306.] EQUIVALENT TERMS. I37 

Now, if everything = AG | AD | BD, then the combinations 
containing AB, ab, AbCD, Abed, aBC, aBcd are inconsistent 
and we eliminate them by making a figure 1 in those sections. 

We can now read in the Reasoning Frame, 

(1) Everything = AC | AD | BD 

(2) Nothing = AB | ab | AbCD | Abed | aBC | aBcd 

(3) Nothing = CD | cd | ABCd | ABcD j abCd | abcD 
The result proves that (2) and (3) are equivalent to (1). 

304. By the Law of Contradiction such an expression as 
Bb has no meaning and, therefore, such a phrase as A or Bb, 
simply means A. We can reject the Bb as surplusage. 

305. By the Law of the Excluded Middle we can always 
affirm that A is AB or Ab, and consequently the phrase AB or 
Ab, means simply A. And again, A (B or b), means nothing 
but A. After working our problems we shall frequently, in 
reading our Reasoning Frames, find such combinations as AB 
or Ab. We can, in reading, always simplify them by omitting 
the B or b and calling the phrase simply A. 

Whenever in a conclusion we have the equivalent for any let- 
ter, we can always substitute the letter, in reading, for the 
equivalent. 

EXERCISES. 

30G. (1) What is the opposite of BA? 

(2) What is the opposite of Ab or aB? 

(3) What is the opposite of ba? 

(4) What is the opposite of A or Be? 

(5) What is the opposite of A or B or cD? 

(6) What is a complete equivalent for Ab or Ac? 

(7) What is a complete equivalent for CAbd or DAbc 

or DBca? 



CHATTER XI. 

ELIMINATION. 

307. Elimination may be considered as a process for getting 
rid of inconsistent propositions. 

Given any proposition, we take letters to stand for the terms, 
and then we combine the letters by means of our Reasoning 
Frame, into all the possible combinations which can be made of 
them. Each one of these combinations represents a bundle of 
propositions. 

308. When we eliminate a combination because it is incon- 
sistent with our premises, we mean thai the bundle of propo- 
sitions contained in the eliminated combination is inconsistent 
or irrelevant for our present purposes. It does not follow that 
the combination is inconsistent with other premises, — the same 
combinations might in other rases be consistent which in the 
case before us are inconsistent. 

309. When we have made all our possible combinations and 
before we have eliminated any of them, the various combina- 
tions have nothing to say as to whether they are true or false. 
Those that are true and those that are false depend upon the 
results of the eliminating process. After we have completed 
the eliminating process, the propositions remaining in the 
Frame are taken to be true because they are consistent with 
the premises and the premises are supposed to be true, and the 
propositions which are eliminated because they are inconsist- 
ent with the premises are supposed to be false because the 
premises are taken to be true. 

310. But, after all, a supposition always lies behind a pre- 
mise. When we say A is B, we really mean, If A is B, and 
hence we can say If A is B, such and such will be our conclu- 
sions. Absolute knowledge is not given to any mortal being. 



§ 311.] CERTAINTY. 130 

311. Every premise will cause us to eliminate certain combi- 
nations (that is, propositions), because they are inconsistent,, 
and we can say with certainty this much, that if our premise, 
for instance, AB, is true, surely the combination Ab is false in 
this case, because by the Law of Contradiction a thing cannot 
both be and not be at the same time, and if this is not certainty, 
then there is no such thing as logic. If A is B we can feel sure 
that there is no thing which has both names A and b. And, 
similarly, if A is B we can feel sure that no thing has the names* 
a and B. 



CHAPTER XII. 

EXAMPLES CONTAINING THREE TERMS. 

312. In order to illustrate the practical application of our 
system to logical problems containing three terms, it will be ap- 
propriate for us to work out a number of examples taken from 
other writers. 

Granite is not a sedimentary rock; 
Basalt is not a sedimentary rock. 
Let. A = granite, 

b = not-a-sedimentary rock, 
C = Basalt. 
The premises can be stated: 

(1) A = Ab 

(2) G = Cb 

We repeat the subject in the predicate in order to have our 
symbolical propositions true when read both ways, and we say 
C is Cb because the order of the letters makes no difference — 
Cb means exactly the same thing as bC. 

Make an ABC diagram: 



AB 


Ab 


aB 


ab 




J 




2 




C 


1 








c 



Fig. 78. 

Now, if A = Ab, then the combinations ABC, ABc, are 
inconsistent, because they imply that A = AB, and we there- 
fore eliminate them by making a figure 1 in those sections. 

If c = Cb, then the combinations ABC, aBC, are inconsist- 
ent because they imply that C = CB and we therefore elimi- 
nate them by making a figure 2 in those sections. 



§ 313.] 



EXAMPLES. 



141 



From the combinations which remain we can obtain the fol- 
lowing definitions: 

(1) A = AbC | Abe, which interpreted is, Granite is not a 

sedimentary rock and is either Basalt or not Basalt. 

(2) C = CAb | Cab, which translated is, Basalt is not a 

sedimentary rock and is granite or not granite. 
(3 B = Bac, which translated is, 

A sedimentary rock is not Granite and is not Basalt. 
(4) a = aBc | abC | abc, which translated is, 
Whatever is not Granite is either a sedimentary rock and 
not-Basalt, or not a sedimentary rock and is Basalt, or is 
neither a sedimentary rock nor Basalt. 

We might go on and give the definitions of all the other 
single terms and of all the compound terms which are to be 
found remaining in the diagram, but it is not necessary. This 
shows us how easy it is with one operation, to obtain all the 
information which is latent in the premises. 

313. This example and several others which follow, are taken 
from Prof. Jevons. 

All planets are subject to gravity; 
Fixed stars are not planets. 
Let A = planets, 
B = fixed stars, 
C = gravity. 
The propositions can be stated, 
A = AC 
B = aB 
Make an ABC diagram, 



AB 


Ab 


aB 


ab 


C 


2 




2 

1 


1 






c 



Fig. 79. 



142 EXAMPLES CONTAINING THREE TERMS. [Chap. 12. 

Now, if A = AC, then the combinations ABc, Abe are incon- 
sistent because they imply that A = Ac, and we therefore 
eliminate them by making a figure 1 in those sections. 

Again, if B = aB, then the combinations ABC and ABc are 
inconsistent because they imply that B = AB, and we there- 
fore eliminate them by making a figure 2 in those sections. 

From the propositions which remain in the Reasoning Frame 
we can get the following definitions: 
(1) A = AbC, which translated is. 

Planets are not fixed stars and are subject to gravity. 
{2) B = aBC | aBc, which interpreted is, 

Fixed stars are not planets and they are or are not sub 
ject to gravity. 
(3) C = AbC ! aBC | abC, which translated is, 

Whatever is subject to gravity is either a planet, and in 
that case it is not a fixed star or it is a fixed star, and 
then it is not a planet, or it is something which is 
neither a planet nor a fixed star. 
We can simplify the last by saying, whatever is subject to 
gravity is either a planet, or a fixed star, or neither. Some- 
times it requires considerable ingenuity to translate our com- 
binations into popular English. 

314. He that is of God, heareth my words: Ye therefore 
hear them not because ye are not of God. 
Let A = he that is of God, 

B — heareth my words, 

C = ye 

a = are not of God, 

b = hear them not. 
The premises can be stated: 

(1) A = B 

(2) B = A 

(3) C = Ca 
Make an ABC diagram: 



§ 315.] 



EXAMPLES. 



143 



AB 


Ab 


aB 


ab 




3 


31 


2 




C 




1 


2 




c 



Fig. 80. 

Now, if A = B, then the combinations AbC, Abe are incon- 
sistent, because they imply that A = b, and we therefore elim- 
inate them by making a figure 1 in those sections. 

Again, if B = A, then the combinations aBC, aBc are incon- 
sistent because they imply that B = a, and we therefore elim- 
inate them by making a figure 2 in those sections. 

Again, if C — Ca, then the combinations ABC and AbC are 
inconsistent because they imply that C = A, and we therefore 
eliminate them by making a figure 3 in those sections. 

From the combinations which remain we can obtain the fol- 
lowing definitions : 

(1) A == ABc, which translated is: 

He that is of God heareth my words and is not ye. 

(2) B = BAc, which translated is: 

He that heareth my words is of God and is not ye. 

(3) C = Cab, which translated is: 

Ye are not of God and ye do not hear my words. 

315. John is a man, 

John is not a triangle. 

Let A = John, 
B = man, 
C = triangle. 

We can state the propositions: 

(1) A =r AB 

(2) A = Ac 

Make an ABC diagram: 



144 



EXAMPLES CONTAINING THREE TERMS. [ Chap. 12. 



AB 


Ab 


aB 


ab 




2 


2 1 






C 




1 






c 



Fiff. 81. 



Now, if A = AB,then the combinations AbC, Abe are incon- 
sistent, because they imply that A = b, and we therefore 
eliminate them by making a figure 1 in those sections. 

Again, if A — Ac, then the combinations ABC, AbC are 
inconsistent because they imply that A = C, and we there- 
fore eliminate them by making a figure 2 in those sections. 

From the combinations which remain in the Reasoning Frame 
we can obtain the following definitions: 

(1) A = A Be, which translated is: 

John is a man and not a triangle. 

(2) B = aBC | aBc | ABc, which translated is: 

A man is either not-John and is a triangle, or a man is 
not-John and is not a triangle, or a man is John and 
not a triangle. 

We can simplify this by saying that it means: 

A man is either John or a triangle or neither. 

(3) C = aBC | abC, which translated is: 

A triangle is not-John, and is either a man or not a man. 

31(». All avaricious men refuse to give money; 
This man refuses to give money. 

Let A = all avaricious men, 



B 



refuse to give money. 



C — this man. 

The premises can be stated thus: 

(1) A = AB 

(2) C = CB. 



816.] AN EXAMPLE. 

Make an ABC diagram . 



145 



AB 


Ab 


aB 


ab 






21 




2 


C 




1 






c 



Fig. 82. 

Now, if A = AB,then the combinations AbC, Abe are incon- 
sistent, because they imply that A = b, and we therefore 
eliminate them by making a figure 1 in those sections. 

Again, if C = CB, then the combinations AbC, abC are 
inconsistent because they imply that C = b, and we therefore 
eliminate them by making a figure 2 in those sections. 

From the combinations which remain we can obtain the fol- 
lowing definitions : 

(1) C = CAB | CaB, which translated is: 

This man refuses to give money and he is either avari- 
cious or not avaricious. 

(2) AB = ABC | ABc, which interpreted is: 

An avaricious man who refuses to give money is either 
this man, or he is not this man. 

(3) B b BAC | BAc | BaC | Bac, which translated is: 

Those who refuse to give money are either an avaricious 
man who is this man, or an avaricious man who is not 
this man, or this man who is not an avaricious mau, 
or men who are neither avaricious nor this man. 

(4) a = aBC | aBc | abc, which translated is: 

Those who are not avaricious are either those who refuse 
to give money and are this man, or those who refuse 
to give money and are not this man, or those who are 
neither. 

(5) b = abc, which translated is: 

10 



146 



EXAMPLES CONTAINING THREE TERMS. [ Chap. 12. 



Those who do not refuse to give money are neither avari- 
cious nor this man. 

317. A science which furnishes the mind with a multitude 
of useful facts deserves cultivation; 
Logic is not such a science. 

Let A = a science which furnishes the mind with a multi- 
* tude of useful facts; 
B — deserves cultivation. 
C = logic. 

We can state the propositions thus: 

(1) A = AB 

(2) C = Ca. 

Make an ABC diagram: 



AB 


Ab 


aB 


ab 




2 


21 






C 




1 






c 



Fiff. 83. 



Now, if A = A B, then the combinations AbC, Abe are incon- 
sistent because they imply that A = b, and we therefore 
eliminate them by making a figure 1 in those sections. 

Again, if C = Ca then the combinations ABC, AbC are 
inconsistent because they imply that C = CA, and we there- 
fore eliminate them by making a figure 2 in those sections. 

From the combinations which remain we can get the follow- 
ing definitions: 

(1) AB = ABc, which translated is: 

A science which furnishes the mind with a multitude 
of useful facts and which deserves cultivation is not 
/ lo«nc. 



§ 318.] AN EXAMPLE. I47 

(2) B == ABC I aBC | aBc, which translated is: 

Whatever deserves cultivation is either a science 
which furnishes the mind with a multitude of useful 
facts and is logic, or something which is not a 
science which furnishes the mind with a multitude 
of useful facts and which is logic, or something which 
is not a science which furnishes the mind with a multi- 
tude of useful facts and which is not logic. 

(3) C = CaB I Cab, which translated is: 

Logic is not a science which furnishes the mind with a 
multitude of useful facts and it deserves or it does not 
deserve cultivation. 

(4) b = abC I abc, which interpreted is: 

Whatever does not deserve cultivation is not a 
science which furnishes the mind with a multitude of 
useful facts, and it is logic or it is not logic. 

318. Mont Blanc is the highest mountain in Europe; 

The highest mountain in Europe is deeply covered with 
snow. 

Let A — Mont Blanc, 

B = the highest mouutain in Europe, 

C = deeply covered with snow. 
We can state the proposition thus: 

(1) A --=-- B 

(2) B == A 

(3) B =-- BC. 

In this case it is clear that if Mont Blanc is the highest 
mountain in Europe, the highest mountain in Europe is Mont 
Blanc, and as it is necessary that we should always read every 
meaning which is true on the face of a proposition, we there- 
fore read B = A. The reason why we do not rend BG = B, 
is because BC = B would cause us to eliminate exactly the 
same combinations which the proposition B = BC causes 
us to eliminate. It would therefore be unnecessary to read 
BC = B. 



148 EXAMPLES CONTAINING THREE TERMS. [ Chap. 12. 

Make aD ABC diagram: 



AB 


Ab 


hB 


ab 






1 


o 




C 


3 


1 


2 
3 




c 



Pig. 84. 

Now, if A = B, then the combinations AbC, Abe, are incon- 
sistent, because they imply that A = b, and we therefore elim- 
inate them by making a figure 1 in those sections. 

Again, if B = A, then the combinations aBC and aBc are 
inconsistent, because they imply that B = a, and we therefore 
eliminate them by making a figure 2 in those sections. 

Again, if B = BC, then the combinations ABc and aBc are 
inconsistent because they imply that B = c, and we therefore 
eliminate them by making a figure 3 in those sections. 

From the combinations which automatically remain, we can 
obtain the following definitions: 

(1) A = BC, which translated is: 

Mont Blanc is the highest mountain in Europe and is 
deeply covered with snow. 

(2) B = AC, which interpreted is: 

The highest mountain in Europe is Mont Blanc and it 

is deeply covered with snow. 
C = AB | Cab, which translated is: 
Whatever is deeply covered with snow is either Mont 

Blanc, which is the highest mountain in Europe, or it 

is not Mont Blanc, and it is something which is not the 

highest mountain in Europe, 
a = abC | abc, which translated is: 
Whatever is not Mont Blanc is not the highest mountain 

in Europe, and it is deeply covered with snow, or it is 

not deeply covered with snow. 



(3) 



W 



§ 319.] 



AN EXAMPLE. 



149 



(5) c = abc, which translated is: 

What is not deeply covered with snow is not Mont 
Blanc, and is not the highest mountain in Europe. 

319. Sodium is a metal; 

Metals conduct electricity. 
Let A = sodium, 

B — metal, 

C = conducts electricity. 
The propositions may be stated thus: 

(1) A = AB 

(2) B = BO 
Make an ABC diagram: 



AB 


Ab 


aB 


ab 






1 






C 


2 


1 


2 




c 



Fig. 85. 

Now, if A = AB, then the combinations AbC, Abe, are incon- 
sistent, because they imply that A = b, and we therefore elimi- 
nate them by making a figure 1 in those sections. 

Again, if B = BO, then the combinations ABc, aBc are incon- 
sistent, because they imply that B = c, and we therefore elimi- 
nate them by making a figure 2 in those sections. 

From the combinations which remain we can obtain the fol- 
lowing definitions: 

(1) A = ABC, which translated is: 

Sodium is a metal which conducts electricity. 

(2) B = ABC | aBC, which translated is: 

A metal conducts electricity, and it is either sodium or 
not sodium. 

(3) C = ABC | aBC | abC, which translated is: 

Whatever conducts electricity is either sodium or a 
metal, or neither. 



150 



EXAMPLES CONTAINING THREE TERMS. [ Chap. 12. 



(4) c = abc, which translated is: 

Whatever does not conduct electricity is not sodium and 
is not a metal. 
320. Neptune is a planet; 

What has retrograde motion is not a planet. 
This last premise in popular English could be expressed, 
No planet has retrograde motion, or, 
A planet has not retrograde motion. 
Let A = Xeptune, 
B = planet, 
C = retrograde motion. 
The propositions may be stated thus: 

(1) A = AB 

(2) C = Cb 
Make an ABC diagram : 



AB 


Ab 


aB 


ab 




2 


1 


2 




C 




1 






c 



Fig. 86. 

Now, if A = AB, then the combinations AbC, Abc, are incon- 
sistent, because they imply that A = b, and we therefore elimi- 
nate them by making a figure 1 in those sections. 

Again, if C = Cb, then the combinations ABC, aBC are 
inconsistent, because they imply that C = CB, and we there- 
fore eliminate them by making a figure 2 in those sections. 

We can obtain the following definitions from the combina- 
tions which remain: 

(1) A == ABc, which translated is: 

Neptune is a planet which has not retrograde motion. 

(2) B = ABc | aBc, which translated is: 

A planet has not retrograde motion, and it is either 
Neptune or not Neptune. 



§ 321.] 



AN EXAMPLE. 



151 



(3) C = abC, which translated is: 

Whatever has retrograde motion is not Neptune and is 
not a planet. 
321. Whales are not true fish; 

True fish respire water. 
Let A = whale, 
B = true fish, 
C = respire water. 
The propositions may be stated thus: 

(1) A = Ab 

(2) B = BG 

Make an ABC diagram: 



AB 


Ab 


aB 


ab 




1 








C 


12 




2 




c 



Fig. 87. 

Now, if A = Ab, then the combinations ABC, ABc, are incon- 
sistent, because they imply that A = AB, and we therefore 
eliminate them by making a figure 1 in those sections. 

Now, if B = BC, then the combinations ABc, aBc, are incon- 
sistent, because they imply that B =* Be, and we therefore 
eliminate them by making a figure 2 in those sections. 

From the combinations which automatically remain we can 
obtain the following definitions: 

(1) A = AbC | Abe, which translated is: 

A whale is not a true fish and does or does not respire 
water. 

(2) B = BaC, which translated is: 

A true fish respires water and is not a whale. 

(3) C = CAb | CaB | Cab, which translated is: 

Whatever respires water is or is not a whale, and it is or 
is not a true fish. 



152 



EXAMPLES CONTAINING THREE TERMS. [ Chap. 12. 



(4) c = cAb | cab, which translated is: 

Whatever does not respire water is not a true fish, and 
it is or is not a whale. 
In the old logic this is called a syllogism in the mood 
Camestres. The old logic could not deal with this mood 
directly, it had to deal with it indirectly by reducing it to some 
other form. Similarly with the mood Baroco, which gave the 
old logicians considerable trouble. 

322. The following is an example taken from Prof. Jevons: 
All heated solids give continuous spectra; 
Some nebulae do not give continuous spectra. 
Let A = all heated solids, 

B = give continuous spectra, 

C = some nebulae. 
The propositions may be stated thus: 

(1) A = AB 

(2) Cb = Cb 

It is clear that "some nebulae" referred to in the subject, 
mean the nebulae which do not give continuous spectra; we 
can express this definitely by using the phrase Cb. 

Make an ABC diagram: 



AB' 


Ab 


aB 


ab 






1 






C 




1 






c 



Fig. 88. 

Now, if A = AB, then the combinations AbO, Abe, are 
inconsistent, because they imply that A = Ab, and we therefore 
eliminate them by making a figure 1 in those sections. 

Again, if Cb = Cb, then no combinations are inconsistent 
with Cb = Cb. 

From the combinations which remain we can obtain the fol- 
lowing definitions: 



§ 323] 



AN EXAMPLE. 



153 



(1) A = ABC | ABc, which translated is: 

Heated solids give continuous spectra and are or are not 
nebulae. 

(2) C =CAB | CaB | Cab, which translated is : 

Nebulae are either heated solids giving continuous 
spectra, or not heated solids giving continuous spec- 
tra, or not heated solids not giving continuous spectra. 
The conclusion which the old logic drew in regard to 
"nebulae" was, "Some nebulse are not heated solids." This is 
a popular definition of "nebulae." The strict logical definition 
is the one given bj us. The logical definition says that nebulae 
are either heated solids giving continuous spectra, or not 
heated solids giving continuous spectra, or not heated solids 
not giving continuous spectra. But it does not tell us that 
some nebulae are not heated solids, — that is a matter of fact 
which happens to be true in this case. 

323. All fixed stars are self-luminous; 

Some heavenly bodies are not self-luminous. 
Let A = fixed-stars, 

B a= self-luminous, 

C = heavenly-bodies. 
The premises may be stated, 

(1) A = AB 

(2) Cb = Cb 

Cb = Cb may be translated, "Heavenly bodies which are not 
self-luminous are not self-luminous," — a self-evident propo- 
sition. 

Make an ABC diagram: 



AB 


Ab 


aB 


ab 






1 






C 




1 






c 



Fis. 89. 



154 EXAMPLES CONTAINING THREE TERMS [Chap. 12. 

Now, if A = AB.then the combinations AbC, Abe are incon- 
sistent, because the}' imply that A = Ab, and we therefore 
eliminate them by making a figure 1 in those sections. 

From the combinations which remain we can obtain the fol- 
lowing definitions: 

(1) AB = ABC | ABc, which translated is: 

Fixed stars which are self-luminous are either heavenly- 
bodies or are not-heavenly-bodies. 

(2) aB = aBC | aBc, which translated is: 

Things which are self-luminous and not-fixed-stars are 
either heavenly-bodies or not-heavenly-bodies. 

(3) C = CAB | CaB | Cab, which translated is: 

Heavenly bodies are either fixed-stars and self-luminous, 
or they are self-luminous but not-fixed-stars, or they 
are neither fixed-stars nor self-luminous. 

The conclusion which the old logic drew was, "Some heav- 
enly bodies are not fixed stars.'' 

Trof. Jevon's theory that logic is the substitution of similars, 
is a half truth. Logic is the art of finding equivalents and con- 
tradictories, consistent and inconsistent propositions. Of 
course, when we have found equivalents, then we can substi- 
tute one for the other. 

324. Some metals are of less density than water; 

All bodies of less density than water will float upon the 
surface of the water. 
Let A = metals, 

B = bodies of less density than water, 
C = float upon the surface of the water. 
The propositions may be stated: 

(1) AB == AB 

(2) B = C 

(3) C = B 

When we have the word "some'' in the subject and predi- 
cate, we can state the proposition in the form of a self-evident 
proposition. 

Make an ABC diagram: 



§324.] 



AN EXAMPLE. 



155 



AB 


Ab 


aB 


ab 






2 




2 


C 


1 




1 




c 



Fig. 90. 



Now, if AB = AB, there are no combinations in the Frame 
which are inconsistent with it, and there is nothing to elimi- 
nate. The only object of stating a proposition which is in the 
form of AB = AB, is simply to posit the terms. 

It means, simply, that there are, in this case, things which we 
call A and B in the Universe of Discourse. When we posit A, 
of course A affirms its own existence, but it does not necessarily 
deny the existence of anything else. 

AB = AB means that there are such things in the Universe 
of Discourse as AB, that is, things which have the names A 
andB. 

Again, if B = C, then the combinations ABc and aBc are 
inconsistent because they imply that B = c, and we therefore 
eliminate them by making a figure 1 in those sections. 

We stated a third proposition, viz.: = B, because it is 
evident that whatever will float upon the surface of water is 
of less density than water, and in stating our propositions we 
should always remember to state every proposition which is, 
prima facie, true, and which can be gathered from the given 
propositions. 

Now, if C = B, then the combinations AbC, abC, are incon- 
sistent, because they imply that C = b, and we therefore 
eliminate them by making a figure 2 in those sections. 

From the combinations which remain we can obtain the fol- 
lowing definitions: 

(1) A = ABC | Abe, which translated is: 

Metals are either of less density than water and will 
float upon the surface of water, or they are not of less 



156 EXAMPLES CONTAINING THREE TERMS. [Chap. 12. 

density than water and will not float upon the surface 
of water. 

(2) B = ABC | aBC, which translated is: 

Whatever is of less density than water is a metal or not a 
metal, and it will float upon the surface of water. 

(3) C = ABC | aBC, which translated is: 

Whatever will float upon the surface of water is a metal 
or not a metal and it is of less density than water. 

(4) a = aBC | abc, which translated is: 

Whatever is not a metal is either of less density than 
water and will float upon the surface of water, or it 
is of not less density than water and will not float 
upon the surface of water. 

(5) b = Abc | abc, which translated is: 

Whatever is not of less density than water is either a 
metal and will not float upon the surface of water, or it 
is not a metal and will not float upon the surface of 
water. 

(6) c = Abc | abc, which translated is: 

Whatever will not float upon the surface of the water is 
not of less density than water and is either a metal or 
is not a metal. 
The conclusion that the old logic drew was, "Some metals 
will float upon the surface of water." 

EXERCISES. 

325. What conclusions can be drawn from the following 
pairs of premises? 

(1) A = AB 
C = Cb 

(2) b = ba 
c = ca 

(3) a = aB 
c = cA 

(4) a = ab 

b = bAc 

(5) AB = AB 



§ 325.] EXERCISES. 157 

C = Ca 

(6) b = ba 
c = cA 

(7) B = Ba 

c = cd 

(8)aB = aB 

b = ba 



CHAPTER XIII. 

DISJUNCTIVES. 

326. In this chapter we shall consider Disjunctive propo- 
sitions. Disjunctive terms are separated by the little conjunc- 
tion "or," our sign for which is a short perpendicular mark, 
thus: |. 

327. A Disjunctive proposition implies opposition, incom- 
patibility, separation. 

328. The Law of Identity holds good for Disjunctive proposi- 
tions. The Law of Identity says that A = A. It is also true 
that A or B = A or B, and as the order of the terms makes no 
difference, it is also true that B or A = A or B. 

329. Prof. Jevons in his Principles of Science says: "Few or 
no logicians except De Morgan, have adequately noticed the 
close relation between combined and disjunctive terms, viz: 
that every disjunctive term is the negative of a corresponding 
combined term and vice versa." Consider the term, 

Malleable dense metal. 

How shall we describe the class of things which are not mal- 
eable dense metals? Whatever is included under that term 
must have all the qualities of malleability, denseness and metal- 
licity. Wherever one or more of the qualities is wanting, the 
combined term will not apply, hence the negative of the whole 
term is: "Not-malleable or not-dense or not-metallic." 

Let A = malleability, 

B = denseness, 

C = metallicity. 
What is the negative of ABC? 

Make an ABC diagram: 



§ 330.] 



COMBINED AND DISJUNCTIVE TERMS. 



159 



AB 


Ab 


aB 


ab 






1 


1 


1 


C 


1 


1 


1 


1 


c 



Fig. 91. 

Let us assume that ABC is the only combination in an ABC 
Reasoning Frame. 

Now, if the only combination is ABC, then the combinations 
containing ABc, Ab, a are inconsistent and we eliminate them 
by making a figure 1 in those sections. 

We can now read in the Reasoning Frame: 

(1). The only combination is ABC. 

(2) No combination is ABc | AbC | Abe | aBC | aBc | 

abC | abc. 
Which can be reduced to: 



Nothing 



ABc I Ab I a, hence, ABc I Ab I a is the 



negative of ABC. 

330. Prof. Jevons further says on the same page, 72, : "In 
the above, (i. e. not-malleable, not-dense or not-metallic) the con- 
junction "or" must necessarily be interpretated as unexclusive, 
for there may readily be objects which are both not-malleable 
and not-dense and perhaps not-metallic at the same time. If 
in fact, we were required to use "or" in a strictly exclusive man- 
ner, it would be requisite to specify seven distinct alternatives 
in order to describe the negative of a combination of three 
terms. The negative of four or five terms would consist of 
fifteen or thirty-one alternatives. This consideration alone, is 
sufficient to prove that the meaning of "or" cannot be always 
exclusive in common language," 

Now, in the example which he has given us it is requisite to 
specify seven distinct alternatives in order to describe tbe nega- 
tive of a combination of three terms. 



160 DISJUNCTIVES. [ Chap. 13. 

331. Prof. Venn in his Symbolic Logic, p. 280, in treating of 
this same subject, says: "The full contradictory of any given 
class expression may be defined as comprising 'all the rest' 
required to make up the Universe with which we are concerned. 
Thus, in the simplest case X and x are contradictories because 
X and x = 1. The complete theoretic process, therefore, for 
assigning the contradictory for any class expression involving 
two, three, four, etc., terms, would be to develop unity into its 
four, eight, sixteen, etc., elements, and then subtract from this 
the given expression. The remainder is the contradiction 
required." 

332. An examination of any Reasoning Frame will convince 
the reader that it is impossible to eliminate all the combinations 
but one, without removing one or more letters, and, of course, 
as we have repeatedly shown, when we eliminate one letter 
entirely, we eliminate every combination in the Frame. Now, 
the logical meaning of eliminating every combination in the 
Reasoning Frame is that our premises are inconsistent. Every- 
thing = ABC is an impossible proposition. 

333. The old logic took little notice of many forms of dis- 
junctive propositions. 

The disjunctive proposition may have a disjunctive subject 
or a disjunctive predicate, or both. 

Prof. Jevons gives this proposition as an example of the 
doubly disjunctive form, 

Solids or liquids or gases are electrics or conductors of elec- 
tricity. 

Let A = solids, 
B = liquids, 
C = gases, 
D = electrics, 

E = conductors of electricity. 

Now, let us make an ABCDE diagram : To make an ABODE 

diagram, we first make an ABCD diagram and then we divide 

the CD sections into those which are E and which are e, by 

drawing a horizontal line through the center of the CD sec- 



§ 333.] 



AN EXAMPLE. 



161 



tions, which gives us the CDE and CDe sections; and by divid- 
ing the Cd sections into those which are E and those which are 
e, by drawing a horizontal line through the Cd sections, which 
gives us the CdE and the Cde sections, and by dividing the cD 
sections into those which are E and those which are e, by draw- 
ing a horizontal line through the cD sections, which gives us t he 
cDE and cDe sections; and bydividing the cd sections into those 
which are E and e, by drawing a horizontal line through the cd 
sections, which gives us the cdE and the cde sections, thus: 



AB 


Ab 


aB 


ab 












CDE 










CDe 










CdE 










Cde 










cDE 










cDe 










cdE 










cde 



Fig. 92. 

I assume that we can read the proposition backward, thus: 

Electrics or conductors of electricity are solids or liquids or 
gases. 

The propositions can be stated thus: 

(1) A | B | C = D | E 

(2) D | E = A | B | 

Make an ABCDE diagram: 



11 



163 



DISJUNCTIVES. 



[Chap. 13. 



AB 


Ab 


aB 


ab 










1 


CDE 


2 


2 


9 




CDe 


2 


2 


2 




CdE 








1 


Cde 




1 


1 




cDE 


2 






2 


cDe 


2 






2 


cdE 




1 


1 




cde 



Fig. 93. 



The full logical expression of the proposition A | B | C = 
D | E is: 

Abe | aBc | abC = De | dE. 

Now, if A | B | C = D | E, then the combinations con- 
taining AbcDE, Abcde, aBcDE, aBcde, abCDE, abCde, are in- 
consistent and we eliminate them by making a figure 1 in those 
sections. 

Again, if D | E = A | B | C, then the combinations con- 
taining ABCDe, ABCdE, ABcDe, ABcdE, AbCDe, AbCdE, 
aBCDe, aBCdE, abcDe, abcdE, are inconsistent and we elimini- 
nate them by making a figure 2 in those sections. 

From the combinations which remain we can get the follow- 
ing definitions: 



(1) ABC 


= DE 


| de 


(2) ABc 


== DE 


| de 


(3) AbC 


== DE 


| de 


(4) Abe 


= De | 


dE 


(5) aBC 


= DE 


| de 


(6) aBc 


= De | 


dE 


(7) abC 


= De | 


dE 


(8) abc 


= DE 


| de 



§334.] 



AN EXAMPLE. 



1G3 



They can be translated thus: 

(1) What is solid and liquid and gaseous is either both an 
electric and conductor of electricity or neither. 

(2) What is solid and liquid is either both an electric and 
conductor of electricity or neither. 

(3) What is both solid and gaseous is either both an electric 
and conductor of electricity or neither. 

(4) A solid is either an electric or a conductor of electricity. 

(5) What is both liquid and gaseous is either both an elec- 
tric and conductor of electricity or neither. 

(6) A liquid is either an electric or a conductor of electricity. 

(7) A gas is either an electric or a conductor of electricity. 

(8) What is neither solid nor liquid nor gaseous is either 
both an electric and conductor of electricity or neither. 

We can also get the following definitions : 

(1) DE | de =AC | aBC | ABc | abc, 

(2) De | dE = Abc | aBc | abC. 
They can be translated thus : 

(1) What is both an electric and conductor of electricity or 
neither, is solid and gaseous or liquid and gaseous or solid and 
liquid or neither solid nor liquid nor gaseous. 

(2) What is an electric or a conductor of electricity is a solid 
or a liquid or a gas. 

334. Given the disjunctive proposition, A | B = C | D, the 
question is, what propositions are equivalent to it? 

Make an ABCD diagram: 



AB 


Ab 


aB 


ab 






1 


1 




CD 










Cd 










cD 




1 


1 




cd 



Fig. 94. 



164 



DISJUNCTIVES. 



[Chap. 13. 



Now, if A | B = C | D, then the combinations containing 
AbCD, Abed, aBCD, aBcd, are inconsistent and we eliminate 
them by making a figure 1 in those sections. 

We can now read in the Reasoning Frame the following 
equivalent propositions. 

| AbCd 

aBCd | 

BaCd 

bAcD I 



AbcD, 



abcD, 



BacD, 

bACd. 



(1) A = AB 

(2) a = ab | 

(3) B = BA 

(4) b = ba | 
Let us assume that the proposition, 

A | B = C | D can be read backward, 
The two propositions can be stated thus: 

(1) A | B = C | D 

(2) C | D = A | B 

Make an ABCD diagram: 



AB 


Ab 


aB 


ab 






1 


1 




CD 


2 






o 


Cd 


o 






2 


cD 




1 


1 




cd 



Fig. 95. 

Now, if A | B = C | D, then the combinations containing 
AbCD, Abed, aBCD, aBcd, are inconsistent and we eliminate 
them by making a figure 1 in those sections. 

Again, if C | D = A | B, then the combinations containing 
ABCd, ABcD, abCd, abcD, are inconsistent and we eliminate 
them by making a figure 2 in those sections. 

From the combinations which remain we can get the follow- 
ing definitions by translating the figures in the sections by the 

word "No." 

(1) No A | B = CD | cd 

No C I D = AB I ab 



§§ 335, 336.] EQUIVALENTS. 

(2) No BD | bd = Ac | aC 
No AC | ac = B | D 

(3) No AD | ad = B | 
No BO I be = A I D 



1G5 



335. Given the proposition ab 
propositions are equivalent? 

Make an ABCD diagram. 



cd, the question is, what 



AB 


Ab 


aB 


ab 










1 


CD 








1 


Cd 








1 


cD 










cd 



Fig. 96. 

•» 
Now, if ab = cd,then the combinations containing abC, abcD, 

are inconsistent and we eliminate them by making a figure 1 in 

those sections. 

We can now read in the Seasoning Frame, the following 
propositions, which taken together are equivalent to the given 
proposition. 

(1) CD == ACD | aBCD 

(2) Cd = CdA | CdaB 

(3) cD == cDA | cDaB 

336. Let us assume that the proposition, ab = cd, can be 
read backward. 

The two propositions can be stated thus: 

(1) ab = od 

(2) cd = ab 

Make an ABCD diagram: 



166 



DISJUNCTIVES. 



[ Chap. 13. 



AB 


Ab 


aB 


ab 










1 


CD 








1 


Cd 








1 


cD 


2 


- 


o 




cd 



Pig. 97. 

Now, if ab = cd, then the combinations containing: abC, 
abcD, are inconsistent and we eliminate them by making a fig- 
ure 1 in those sections. 

Again, if cd = ab, then the combinations containing Acd, 
aBcd, are inconsistent and we eliminate them by making a fig- 
ure 2 in those sections. 

We can now read in the Reasoning Frame, 

(1) ab = cd 

(2) cd=,ab 

(3) A = AC | AcD 

(4) B = BC | BcD 

The appearance of these Reasoning Frames shows that the 
propositions, 

A | B = C | D, and 

ab = cd, do not correspond to each other. 

337. In our system, after stating the propositions we elimi- 
nate in the Reasoning Frame the inconsistent propositions and 
read the results. 

Equivalent propositions will have the same appearance in 
the Reasoning Frames. 
Non-equivalent propositions will have a different appearance. 

338. According to my view, a proposition resembles a box 
which contains many different articles. When we lift the 
cover of the box we can see what is in the box. So in our sys- 
tem, when we eliminate the inconsistent combinations we can 
see the contents of the given proposition. 



§§ 339-342.] 



CONTRADICTORIES. 



1G7 



339. When we wish to ascertain whether propositions are 
equivalent, all that is necessary is to eliminate the inconsistent 
combinations and if the same combinations have been elimina- 
ted, then the propositions are equivalent. 

340. A proposition is the expression of a thought, and, by 
our system, we make thoughts visible. It then becomes an 
easy matter to read them and to tell exactly the extent to which 
they agree or disagree. 

341. Given the disjunctive propositions, 

A | B = C | D 
C | D = A | B 
the question now arises, What propositions are contradictory 
to the given propositions? 

A complete contradictory will cause us to eliminate the com- 
binations which the given propositions saved and to save the 
combinations which the given propositions eliminated. 

342. By the use of the Seasoning Frame I have discovered 
an easy method of solving this problem. It is this: 

From the eliminated combinations get a pair of propositions 
containing the definitions of a letter-term and its negative. 
Any letter-term will do. This pair of definitions will together 
make a full contradictory to the given propositions. 

Make an ABCD diagram: 



AB 


Ab 


aB 


ab 






1 


1 




(D 


2 






2 


Cd 


2 






2 


cD 




1 


1 




cd 



Fig. OS. 



Now, if A | B = C | D, then the combinations containing 
AbCD, Abed, aBCD, aBcd, are inconsistent, and we eliminate 
them by making a figure 1 in those sections. 



1G8 



DISJUNCTIVES. 



[Chap. 13. 



Again, if C | D = A | B, then the combinations ABCd, 
ABcD, abCd, abcD are inconsistent, and we eliminate them by 
making a figure 2 in those sections. 

From the eliminated combinations we can get the following 
pair of definitions: 

(1) B = ACd | AcD | aCD | acd 

(2) b = ACD | Acd | aCd | acD 

Make an ABCD diagram: 



AB 


Ab 


aB 


ab 




1 






•> 


CD 




a 


1 




CI 




~ 


1 




cD 


1 






~ 


cd 



Pie. 91). 



Now, if B = ABCd | ABcD | aBCD | aBcd, then the com- 
binations containing ABCD | ABcd | aBCd | aBcD, are in- 
consistent and we eliminate them by making a figure 1 in 
those sections. 

Again, if b = AbCD | Abed | abCd | abcD, then the com- 
binations containing AbCd, AbcD, abCD, abed are inconsis- 
tent, and we eliminate them by making a figure 2 in those 
sections. 

The result proves that we have found a pair of propositions 
which are completely contradictory to the given propositions. 

We can also reau in tne eliminated combinations the fol- 
lowing pairs of propositions which are each equivalent to 
ihe given propositions. 

(1) No BD | bd == A | C, and 
No AC | ac = B | D. 

(2) No CD | cd = A | B, and 
No AB I ab = C I D. 



313.] 



CONTRADICTORIES. 

(3) No B | D = AC | ac, and 

No A | C = BD | bd. 

(4) No AD | ad = B | C, and 
No BO I be = A I D. 



1G9 



We will prove tne last one. 
Now, if No AD I ad = B 



C, then the combinations con- 



taining ABcD, AbCD, aBcd, abCd, are inconsistent, and we 
eliminate them by making a figure 1 in those sections. 
Make an ABCD diagram: 



AB 


Ab 


aB 


ab 






1 


2 




CD 


2 






1 


Cd 


1 






2 


cD 




2 


1 




cd 



Fig. 100. 

Again, if No BC | be = A | D, then the combinations ABCd, 
aBCD, Abed, abcD are inconsistent, and we eliminate them by 
making a figure 2 in those sections. 

The result proves that the pair of propositions, 
No AD | ad = B | C, and 
No BC | be = A | D, 
is equivalent to the given propositions, 
A | B = C | D 
C | D = A | B 

343. The next question is, Given any alternative definition 
of a letter-term, can we reduce it to a set of categorical propo- 
sitions which shall be equivalent to the given alternative propo- 
sition? 

We can generally do so, and perhaps always. 

From the last diagram we can get the following alternative 
definition of A: 

A = ABCD I ABcd I AbCd I AbcD. 



170 



DISJUNCTIVES. 



[ Chap. 13. 



Wo can reduce this to the following equivalent categorical 

propositions: 

(1) ABC = ABCD 

(2) A Be = ABcd 

(3) AbC = AbCd 

(4) Abe = AbcD 

This quartet of categorical definitions is equivalent to the 
given alternative definition. 
Make an ABCU diagram: 



AB 


Ab 


aB 


ab 






3 






1 D 


1 








Cii 


~ 








« 1) 
in! 




4 







Fig. 101. 



Now, if ABC = ABCD, then the combination containing 
ABCd is inconsistent, and we eliminate it by making a figure 1 

in that section. 

Again, if ABc = ABcd, then the combination ABcD is 
inconsistent, and we eliminate it by making a figure 2 in thai 
section. 

Again, if AbC = AbCd, then the combination AbCD is in- 
consistent and we eliminate it by making a figure 3 in that sec- 
tion. 

Again, if Abe == AbcD, then the combination Abed is incon- 
sistent, and we eliminate it by making a figure 4 in that 
section. 

The result proves the equivalence of the quartet of categori- 
cal propositions with the given alternative proposition. 

3U. The reader will understand that a diagram is at first a 
blank diagram, and that we mark it as we proceed with our 
work, so that when he sees it he does not see the diagram as it 



§§ 345-347.] CATEGORICALS. 171 

was first made, but as it is after we have finished our work. It 
will be proper for him to bear in mind, therefore, that before 
commencing work he is supposed to be looking at a blank dia- 
gram, and each section is supposed to be blank until he is told 
to put some mark in it. 

345. Whenever we wish to obtain the alternative definition 
of a term which is common to several categorical propositions, 
all we have to do is to mark these sections in the Reasoning 
Frame which contain the several categorical propositions, and 
then read the definition of the common term by saying that 
it is one or the other of all these combinations which are 
marked with a sign. 

346. And, again, if we wish to break up an alternative defi- 
nition into categorical propositions, all that is necessary to do 
is to mark the sections which contain the several alternants in 
the alternative definition, with a sign, and then from those sec- 
tions which are so marked we read the categorical definitions 
which they imply. 

347. Let us take another example of a disjunctive propo- 
sition from Prof. Jevon's Principles of Science, p. 74: Senior's 
definition of wealth: 

"Wealth is what is transferable, limited in supply and 
either productive of pleasure or preventive of pain." 

Let A = wealth, 

B = transferable, 

C = limited in supply, 

D = productive of pleasure, 

E = preventive of pain. 

The proposition can be stated, 

A = ABC (D | E) 
which when developed becomes, 

A = ABCDe | ABCdE 

Prof. Jevons gives another alternative, viz.: ABCDE, 
because he places a different meaning on the word "or" from 
that which I do. He says that ordinarily " 



172 



DISJUNCTIVES. 



[Chap. 13. 



the other or both, whilst I contend that the usual meaning of 
"or" is one or the other and not both. 

We can also obtain from Senior's definition of wealth this 
proposition: That whatever is transferable and limited in 
supply and productive of pleasure or preventive of pain, is 
wealth. 

It can be stated thus: 

BC (D or E) = ABC (D or E) 
which can be developed into, 

BCDe | BCdE = ABCDe | ABCdE 

Now let us make another ABCDE diagram: 



AB 


Ab 


aB 


ab 




1 


B 






CDE 




5 


3 




CDe 




5 


4 




CdE 


■ 


5 






Cde 


6 


56 






cDE 


6 


56 






cDe 


6 


56 






cdE 


6 


56 






cde 



Fig. 102. 



Now, if A = ABCDe | ABCdE, then the combination 
ABCDE is inconsistent, because it implies that A = both D 
and E, and we therefore eliminate it by making a figure 1 in 
that section. 

Again, if A = ABCDe or ABCdE, then the combination 
ABCde is inconsistent because it implies that A = neither 
D nor E, and we therefore eliminate it by making a figure 2 in 
that section. 



§ 347.] AN EXAMPLE. 173 

Again, if BCDe === A, then the combination aBCDe is incon- 
sistent because it implies that BCDe = a and we therefore 
eliminate it by making a figure 3 in that section. 

Again, if BCdE = A, then the combination aBCdE is incon- 
sistent because it implies that BCdE = a and we therefore 
eliminate it by making a figure 4 in that section. 

It also appears on the face of the proposition that Wealth is 
transferable, which can be stated, 

A = AB 

We say A = AB because it does not follow from "Wealth 
is transferable" that what is transferable is wealth, and there- 
fore we cannot say that B = A. 

It is not necessary that we should always state in our sym- 
bolical language, in the first instance, all the prima facie mean- 
ings of the given proposition. We may not perceive them 
when we first read the given proposition, but if we perceive 
them afterwards, we can still state them and eliminate the 
inconsistent propositions. One great advantage of this 
system is that as our knowledge increases we state our new 
knowledge in our symbolical language and deduce the conse- 
quences which follow. We are not limited to the propositions 
with which we started at first. 

Now, if A = AB, then all the combinations which contain 
Ab are inconsistent because they imply that A = b, and we 
therefore eliminate them by making a figure 5 in those 
sections. 

Again, it appears on the face of the given proposition that 
Wealth is limited in supply. We can state it thus: 

A = AC 

Now, if A = AC then the combinations which contain Ac 
are inconsistent because they imply that A = c, and we there- 
fore eliminate them by making a figure 6 in those sections. 

From the combinations which remain we can get the follow- 
ing definitions: 

(1) A = ABCDe | ABCdE, which translated is: 

Wealth is transferable, limited in supply, and is either 
productive of pleasure or preventive of pain. 



174 



DISJUNCTIVES. 



[ Chap. 13. 



(2) CDe | CdE = AB | ab, which translated is: 

What is limited in supply and productive of pleasure 
but not preventive of pain, or what is limited in sup- 
ply and preventive of pain but not productive of 
pleasure is either wealth and transferable, or it is 
neither. 
We can also get definitions of the other terms in the given 
proposition, but as the definitions would contain so many 
alternatives they would be of little practical use. 

348. The reader will have noticed that the rule for com- 
bining alternatives is simply to combine each alternant of one, 
with each alternant of the other. 

349. Let us take another example of an alternative propo- 
sition from Prof. Jevons: 

"Gems are either rare stones or beautiful stones." 
Let A = gems, 

B = rare stones, 

C = beautiful stones. 
The proposition can be stated thus: 
A == ABc | AbC. 

Make an ABC diagram: 



AB 


Ab 


aB 


ab 




1 








C 




1 






c 



Fig. 103. 



Now, if A = ABc | AbC, then the combinations ABC and 
Abe are inconsistent and we eliminate them by making a figure 
1 in those sections. ABC is inconsistent because it implies 
that A = both B and 0, and Abe is inconsistent because it 
implies that A = neither B nor C. 



§ 349.] 



AN EXAMPLE. 



175 



From the combinations which remain we can get the follow- 
ing definitions: 

(1) Ac == BAc, which translated is: 

A gem which is not a beautiful stone is rare. 

(2) AC =bAC, which translated is: 

A gem which is a beautiful stone is not rare. 
As a matter of fact, these definitions are not true. It there- 
fore follows that our original proposition was not true. 

The original proposition should have been: 

"Gems are either rare or beautiful stones, or both." 
Our proposition can be stated thus : 

A =± ABc | AbC | ABO 
Make an ABC diagram: 



AB 


Ab 


aB 


ab 












C 




1 






c 



Fig. 104. 

Now, if A = ABc | AbO ABC then the combination 
Abe is inconsistent, because it means that A = neither B nor C, 
and we therefore eliminate it by making a figure 1 in that 
section. 

From the combinations which remain we can get the follow- 
ing definitions: 

(1) A = ABc | AbC | ABC, which translated is: 
Gems are either rare but not beautiful stones or beautiful 

but not rare stones, or they are both beautiful and rare. 

(2) B = AB | aBC | aBc, which translated is: 

A rare stone is either a gem or it is beautiful and not a 
gem, or it is neither. 
The point I wish to make in giving this illustration is, that 
when a man means one or the other or both, he should say so, 



176 DISJUNCTIVES. [ Chap. 13. 

or, if when he says one or the other he clearly means "one or 
the other or both," then we should state what he means and 
not what he says, otherwise our conclusions will be wrong. 

350. Let us take another example from Prof. Jevons: 

(1) Red colored metal is either copper or gold. 

(2) Copper is dissolved by nitric acid. 

(3) This specimen is red colored metal. 

(4) This specimen is not dissolved by nitric acid. 
Let A = this specimen, 

B = red colored metal, 
G = copper, 
D = gold, 

E = dissolved by nitric acid. 
The propositions can be stated as follows: 

(1) B = BCd | BcD 

(2) C = CE 

(3) A = AB 

(4) A = Ae 

(5) C = Cd 
(G) D = Dc 

Now make an ABCDE diagram: 



§ 350.] 



AN EXAMPLE. 



177 



AB 


Ab 


aB 


ab 




45 

1 


54 
3 


5 

1 


5 


CDE 


5 
2 

1 


5 
2 
3 


2 
1 
5 


2 
5 


CDe 


4 


4 
3 






CdE 


2 


3 
2 


2 


2 


Cde 


4 


3 

4 






cDE 




3 






cDe 


4 

1 


4 
3 


1 




cdE 


1 


3 


1 




cde 



Fig. 105. 



Now, if B = BCd BcD then the combinations which 
contain BCD Bed are inconsis tent because they imply tha t 
B = CD | cd, and we therefore eliminate them by making 
a figure 1 in those sections. 

They are ABCDE, ABCDe, ABcdE, ABcde, aBCDE, aBCDe, 
aBcdE, aBcde. 

Again, if C = CE, then all the combinations containing Ce 
are inconsistent because they imply that C = e, and we there- 
fore eliminate them by making a figure 2 in those sections. 

They are ABCDe, ABCde, AbCDe, AbCde, aBCDe, aBCde, 
abCDe, abCde. 

Now again, if A = AB, then all the combinations contain- 

12 



178 DISJUNCTIVES. [Chap. 13. 

ing Ab are inconsistent because they imply that A = b and 
we therefore eliminate them by making a figure 3 in those 
sections. 

They are AbCDE, AbCDe, AbCdE, AbCde, AbcDE, AbcDe, 
AbedE, Abcde. 

Now again, if A = Ae, then all the AE combinations are 
inconsistent because they imply that A =E, and we there- 
fore eliminate them by making a figure 4 in those sections. 

They are ABCDE, ABCdE, ABcDE, ABcdE, AbCDE, 
AbCdE, AbcDE, AbcdE. 

Again, if C = Cd, then all the combinations which contain 
CD are inconsistent because they imply that C = D, and we 
therefore eliminate them by making a figure 5 in those 
sections. 

They are ABCDE, AbCDE, aBCDE,abCDE,ABCDe, AbCDe. 
aBCDe, abCDe. 

Again, if D = Dc, then all the combinations which contain 
CD are inconsistent, but it is unnecessary to eliminate them 
because they are already eliminated. 

From the combinations which remain we can get the follow- 
ing definition: 

A = ABcDe, which translated is: 

This specimen is red colored metal, and not-copper, but 

is gold and is not dissolved by nitric acid: 
which can be contracted into, 
This specimen is gold. 

351. This process of reasoning is called obcissio infiniti, and 
it means the obtaining of a name by getting rid of the alter- 
natives. 

The reader will understand that the reason why so many 
sections in this diagram are not eliminated is, that they are 
not inconsistent with any information given to us in the orig- 
inal propositions. So far as data has been furnished us they 
are true propositions, but of course the definition of a or 
b or C or D or E or c or d or e, would contain so many 
alternants that the definitions would be of little practical use. 



§ 352.] 



AN EXAMPLE. 



179 



As we have stated elsewhere, whenever there is more than 
one combination containing a given letter, then the definition 
of that letter is always an alternative definition. The number 
of alternants in the alternative will be equal to the number of 
combinations which contain the letter which is to be defined. 
352. Let us take the following example from Prof. Venn: 
"The members of a Board were all of them either bond- 
holders or shareholders, but not both, and the bondholders, 
as it happened, were all on the Board." 

Let A = member of the Board, 
B = bondholder 
C = shareholder. 

The propositions can be stated thus: 

(1) A = ABc | AbC 

(2) B = AB 



Make an ABC diagram 



AB 


Ab 


aB 


ab 




1 




2 




C 




1 


2 




c 



Fig. 106. 

Now, if A = ABc AbC, then all the other combinations 

of A are inconsistent. 

There are four A combinations and we therefore eliminate 
the combinations ABC and Abe. 

Again, if B = AB, then all the combinations which contain 
aB are inconsistent, because they imply that B = a and we 
therefore eliminate them by making a figure 2 in those sec- 
tions, which are aBC and aBc. 

From the combinations which remain we can get the follow- 
ing definitions: 



180 DISJUNCTIVES. [ Chap. 13. 

(1) C = CAb | Cab, which translated is: 
Shareholders are either members of the Board and not- 

bondholders, or they are neither members nor bond- 
holders, 
which can be simplified, 
"The shareholders are not bondholders." 
We already know that by the Law of the Excluded Middle, 
that shareholders air either members of the Board or not 
members of the Board, and therefore we can drop this state- 
ment : 

(2) B = ABc, which translated is: 

A bondholder is a member of the Board and not a share- 
holder. 

(3) a =ab (C c), which translated is: 

A non-member of the Board is not a bondholder. 
It is unnecessary to translate Core; we know by the Law 
of the Excluded Middle that anything is C or c. 

(4) c = cAB | cab, which translated is: 

A non-shareholder is either a member of the Board and 
a bondholder, or he is neither. 

Prof. Venn says that "this problem was proposed in examin- 
ation and lecture rooms to some 150 students, as a problem in 
ordinary logic; it was answered by, at most, five or six of them. 
It was afterwards set as an example of Boole's Method, to 
a small class who had attended a few lectures on the nature of 
the Symbolic Methods; it was readily answered by half or more 
of their number.'' 

The conclusion arrived at by Boole's method was, "No share- 
holders are bondholders/' By our method we not only get that 
conclusion, but we get a great many more. 

353. A Disjunctive proposition always implies that one of 
its alternants must be true, otherwise the whole proposition 
must be false. 

354. I want to add a caution to the reader at this point, 
in regard to reasoning from Disjunctive propositions. The 
difficulty is, that in many cases we cannot be sure that the 



§§ 355-357.J AMBIGUITY. 18 l 

alternants are complete, that is, that we have stated all of 
them. When we say A is B or C, the question arises whether 
it may not also be D or E. If we can say that A is only B or C, 
we shall have no trouble. But if it is a doubtful matter 
whether A is only B or C, then we cannot be sure of our con- 
clusions. 

355. Jn working our system we must get rid of every 
ambiguity, for if there is ambiguity in the premises, there will 
be ambiguity in the conclusions, and if there is no ambiguity 
in the premises, there cannot be any ambiguity in the con- 
clusions, if our operations have been correctly performed. 

356. Our system is perfect, but we are liable to make mis- 
takes in working it. First, we may not understand the propo- 
sitions which are given to us and hence we may interpret them 
wrongly. 

Second, through inadvertence, we may omit to eliminate an 
inconsistent combination, or we may eliminate a combination 
which ought to stand. Third, in reading the combinations 
which remain, we may neglect to read one or more of them. 
We should take care not to make these mistakes. 

357. When we are in doubt whether an alternative propo- 
sition contains all the alternants, we should so state the prop- 
osition that it will cover omitted cases, if any such there be. 
Thus, if the proposition given us is, A = B or C, and we 
think that possibly there are other things which "A" may be, 
then we should state it: 

A = ABc | AbO J Abe, 
which translated is : 

A = B | C | neither. 

The proposition as stated thus, simply denies that 
A = both B and C. 

For instance, "Wealth must be either spent or hoarded; it is 
not hoarded, therefore it is spent." This sort of reasoning is 
fallacious, because wealth may be neither spent nor hoarded. 

Again, "If it is spring, you are to blame for not sowing; if 
it is autumn, you are to blame for not reaping, but it is either 



182 



DISJUNCTIVES. 



[Chap. 13. 



spring or autumn, therefore, you are to blame for either not 
sowing or not reaping." This also is fallacious, because the 
enumeration of the parts is not complete — it may be either 
summer or winter. 

358. Where the enumeration is complete and the members 
are exclusive, i. e., only one of them can be true, then if any 
one of them is true, all the others will be inconsistent. 

For instance, "this event occurred in spring, summer, autumn 
or winter; it occurred in spring, therefore it did not occur in 
any of the others." 

And of course it follows that if one of the alternants is not 
true, then one of the others which remain must be true. 
Thus, "It did nut occur in summer, therefore it must have 
occurred in one of the others." 

359. Disjunctive propositions sometimes appear in this 

form : 

Either B or C exists, 

Let A = exists. 

The proposition can be stated thus: 

Be | bC = A 

Make an ABC diagram : 



AB 


Ab 


aB 


ab 










1 


C 






1 




c 



Fig. 107. 



Now, if Be | bC = A, then the combinations aBc, abC are 
inconsistent and we eliminate them by making a figure 1 in 
those sections. 

We can now read in the Reasoning Frame: 

B | C = A, which can be translated thus: 
Either B | C = some existing thing. 



§ 360.] 



AN EXAMPLE. 



183 



360. Prof. Bain in "Deductive and Inductive Logic," p. 119, 
gives the following example of a disjunctive proposition: 
"Either the witness is perjured or the prisoner is guilty." 
Let A = witness, 
B = perjured, 
C — prisoner, 
D = guilty. 
The proposition can be stated thus: 

A = B I G = D 

Make an ABCD diagram: 



AB 


Ab 


aB 


ab 




12 








CD 




21 


2 


2 


Cd 




1 






cD 




1 






Cd 



Fig. 108. 

I assume that the proposition can be read backward thus: 

Either the prisoner is guilty or the witness is perjured. 

It can be stated thus: 

= D | A = B 

Now, if A = B, except where C = D, then the combinations 
containing ABCD, AbCd, Abe are inconsistent, and we elimi- 
nate them by making a figure 1 in those sections. 

Again, if C = D, except where A — B, then the combinations 
containing ABCD, AbCd, aBCd, abCd, are inconsistent, and 
we eliminate them by making a figure 2 in those sections. 

From the combinations which remain we can get the follow- 
ing definitions: 

(1) A = B|C = D 

(2) C == D | A = B 

(3) Ab = CD, which can be translated, 



184 



DISJUNCTIVES. 



[ Chap. 13. 



If the witness is not perjured the prisoner is guilty. 

(4) Cd = AB, which can be translated, 
If the prisoner is not guilty the witness is perjured. 

361. Dr. Keynes in his "Formal Logic," p. 314, gives an 
example of a disjunctive proposition and a categorical propo- 
sition, which is called in the old logic, the modus ponendo 
tollens. Thus: 

A = B | C 
A = B, therefore, 
A == not-G 
which can be stated thus : 

A = ABc | AbO 
A = BA 

Make an ABC diagram: 



AB 


Ab 


aB 


ab 




1 


2 






C 




2 

1 






c 



Fig. 109. 



Now, if A = either ABc | AbC, then the combinations ABC, 
Abe, are inconsistent because they imply that A = either both 
B and C, or neither B and C, and we therefore eliminate them 
by making a figure 1 in those sections. 

Again, if A = AB, then the combinations AbC, Abe, are 
inconsistent because they imply that A = b, and we therefore 
eliminate them by making a figure 2 in those sections. 

From the combinations which remain we can get this 

definition, 

A = ABc 
which translated is, 

A = B and c, which can be contracted into, 



|§ 362, 363.] 



MODUS TOLLENDO PONENS. 



1S5 



A = c, because we are not obliged to give any more of 
a definition than suits our purpose. 

362. In this case we have followed our usual interpretation 
of the word "or." If we had followed the interpretation that 
"or" means one or the other or both, the conclusion obtained 
in this case would not be true. 

The definition of A would have been A = C | c. 
The validity of the modus poncndo tollcns depends upon the 
exclusive meaning of the word "or." 

363. Dr. Keynes in his "Formal Logic," p. 312, gives this 
example of disjunctive reasoning, which in the old logic is 
called, modus tollendo ponens: 

"A is either B or G 
"A is not B 
"Therefore A is C." 
We can state the propositions thus: 

(1) A = ABc | AbC 

(2) A = Ab 

Make an ABC diagram: 



AB 


Ab 


aB 


ab 




1 
2 








C 


2 


1 






c 



Fig. 110. 

Now, if A = ABc | AbC, then the combinations ABC, Abo 
are inconsistent because they imply that A = either both H 
and C or neither, therefore we eliminate them by making n 
figure 1 in those sections. 

Again, if A = b, then the combinations ABC, ABc are incon- 
sistent because they imply that A = B. We therefore elimi- 
nate them by making a figure 2 in those sections. 



1SG 



DISJUNCTIVES. 



[Chap. 13. 



From the combinations which remain we can get the fol- 
lowing definition: 

A = AbC, which can be translated, 

A = C 

364. Let ns take the following example: 

(1) Either A = B | C = D, and conversely, 

(2) A = b. 

Make an ABCD diagram: 



AB 


Ab 


aB 


ab 




12 








CD 


2 


1 


1 


1 


Cd 


o 


1 






cD 


o 


1 






cd 



Fig. 111. 

Now. if A = B, except where C = D, and conversely, then 
the combinations containing ABCD, AbCd, Abe, aCd, are 
inconsistent, and we eliminate them by making a figure 1 in 
those sections. 

Again, if A = b, then the combinations containing AB are 

inconsistent, and we eliminate them by making a figure 2 in 
those sections. 

We can now read in the Reasoning Frame, 

= D i 

365. A writer says, "Logicians have not, as a rule, given 
any distinctive recognition to arguments consisting of two 
disjunctive premises and a disjunctive conclusion, and Mr. 

goes so far as to remark that 'both premises of a 

syllogism cannot be disjunctive, since from two assertions as 
indefinite as disjunctive propositions necessarily are, nothing 
can be inferred.' It is, however, clear that this is erroneous." 
And he gives this example (the lettering is mine): 



§ 365.] 



AN EXAMPLE. 



187 



"Either A is not true or B is true; 
"Either C is not true or A is true; 
"Therefore, either C is not true or B is true." 
I assume that the premises can be stated thus: 

(1) A = d | B = E 

(2) C = f | A = D 

Make an ABCDEF diagram: 



ABC 


ABc 


AbC 


Abe 


aBC 


aBc 


abC 


abc 








3 


3 


6 




6 




DEF 


4 




43 


3 










DEf 


2 


2 


3 


3 


6 




6 




DeF 


24 


2 


43 


3 










Def 


15 


1 


5 




6 




6 




dEF 


1 


1 














dEf 


5 




5 




6 




6 




deF 


















def 



Fig. 112. 

An ABCDEF diagram consists of sixty-four sections. It *s 
made by dividing our square by seven equidistant horizontal 
lines and by seven equidistant perpendicular lines. 

Now, if A = d, except where B = E, then the combinations 
containing ABdE are inconsistent, and we eliminate them by 
making a figure 1 in those sections. 

Again, if A = d, except where B = E, then the combinations 
containing ABDe are inconsistent, and we eliminate them by 
making a figure 2 in those sections. 

Again, if A = d, except where B = E, then the combinations 
containing AbD are inconsistent, and we eliminate them by 
making a figure 3 in those sections. 

Again, if C = f, except where A = D, then the combinations 
containing ACDf are inconsistent, and we eliminate them by 
making a figure 4 in those sections. 



188 



DISJUNCTIVES. 



[ Chap. 13. 



Again, if C = f , except where A = D, then the combinations 
containing ACdF are inconsistent, and we eliminate them by 
making a figure 5 in those sections. 

Again, if C = f, except where A = D, then the combinations 
containing aCF are inconsistent, and we eliminate them by 
making a figure 6 in those sections. 

The result proves that the given premises are not incon- 
sistent, and that there are a large number of inferences which 
can be drawn from the given propositions. 

3G6. A writer says, "In a pure, alternative syllogism, both 
of the premises and the conclusions are alternatives, e. g., 

C is D, or A is not B, 

E is F, or C is not D, 
Therefore, E is F, or A is not B." 
The propositions can be stated thus: 

(1) C = CD | A = Ab 

(2) E = EF | C = Cd 
The conclusion can be stated, 

E = EF I A == Ab 



Make an ABODEF diagram: 



ABC 


ABc 


AbC 


Abe 


aBC 


aBc 


abC 


abc 








1 












DEF 


2 


2 


1 
2 


2 


2 


2 


2 


2 


DEf 






1 












DeF 






1 












Def 


1 

2 




2 




1 
2 




1 
2 




dEF 


1 
2 


2 


2 


2 


1 
2 


2 


1 

2 




dEf 


1 








1 




1 




deF 


1 








1 




1 




def 



Fig. 113. 



§ 367.] 



AN EXAMPLE. 



189 



Now, if C = CD, except where A = Ab, then the combina- 
tions containing AbCD, ABCd, aCd, are inconsistent, and we 
eliminate them by making a figure 1 in those sections. 

Again, if E = EF, except where C = Cd, then the combi- 
nations containing CdEF, CDEf, cEf, are inconsistent, and we 
eliminate them by making a figure 2 in those sections. 

We cannot read in the Seasoning Frame the conclusion 
given in the text. 

Make an ABCDEF diagram: 



ABC 


ABc 


AbC 


Abe 


aBC 


aBc 


abC 


abc 








1 


1 


1 








DEF 


1 - 


1 




1 


1 


1 


DEf 






1 












DeF 
Def 




















1 










dEF 
dEf 


1 


1 






1 


1 


1 


1 


















deF 
def 



















Fig. 114. 



Now, if E = EF, except where A = Ab, then the combina- 
tions containing AbEF, ABEf, aEf, are inconsistent, aud we 
eliminate them by making a figure 1 in those sections. 

This Seasoning Frame now shows the logical expression of 
E = EF | A = Ab. 

The difference in the appearance of the two Reasoning 
Frames is very striking, and it shows that the conclusion given 
in the text is not correct. 

367. In working this system, remember that we look i 
what a proposition denies, or, in other words, we seek for the 



190 



DISJUNCTIVES. 



[ Chap. 13. 



propositions which can be eliminated. We do not look for the 
propositions which agree with the given proposition, but we 
look for those which are inconsistent with it. 

368. A writer gives this example of a disjunctive propo- 
sition: "Every blood vessel is either a vein or an artery.' 
Let A = blood vessel, 
B = vein, 
C = artery. 
The proposition may be stated, 

A = ABc | AbC 

Make an ABC diagram: 



AB 


Ab 


aB 


ab 




1 
3 




4 
5 


6 


C 




2 


8 




c 



Fiff. 115. 



Now, if A = ABc | AbC, then the combination ABC is 
inconsistent because it means that A = both B and C, and we 
eliminate it by making a figure 1 in that section. 

Again, if A = ABc | AbC, then the combination Abe is 
inconsistent because it implies that A = neither B nor C, and 
we therefore eliminate it by making a figure 2 in that section. 

From the combinations which remain we can get the follow- 
ing definitions: 

(1) Ab = C, which translated is: 

A blood vessel which is not a vein is an artery. 

(2) AB = ABc, which translated is: 

A blood vessel which is a vein is not an artery. 

(3) C = Ab | aB | ab, which translated is: 

An artery is a blood vessel and not a vein, or a vein and 
not a blood vessel, or it is neither; 



§3C8.] AN EXAMPLE. 191 

Now, this is the logical definition of an artery derivable from 
the proposition which was given to us. It is not stated in the 
proposition which was given to us, that an artery is not a 
vein, but we can assume that an artery is not a vein and state 
it thus: 

C = Cb. 

Now, if C = Cb, then the combination ABC is inconsistent, 
because it implies that C = B, and we therefore eliminate it by 
making a figure 3 in that section. 

Again, if C = Cb, then the combination aBC is inconsistent 
because it implies that C = B, and we therefore eliminate it 
by making a figure 4 in that section. 

The definition of C now is: 

C = CAb | Cab, which translated is: 

An artery is a blood vessel and not a vein, or it is neither. 
This is now the logical definition from the propositions 
which have been given and assumed of an artery. It is not sta- 
ted in the propositions given and assumed that an artery is a 
blood vessel, but we may assume that an artery is a blood ves- 
sel. 

It can be stated thus: 

C == CA 

Now, if C = CA, then the combination aBC is inconsistent 
because it implies that C = a, and we therefore eliminate it 
by making a figure 5 in that section. 

Again, if C = CA, then the combination abC is inconsistent 
because it implies that C = a, and we therefore eliminate it by 
making a figure 6 in that section. 
Our definition of C now is, 
C = CAb, which translated is: 

An artery is a blood vessel and not a vein, or, simplified, 
An artery is a blood vessel. 
It is not stated in the proposition given us or in the propo- 
sition assumed, that a vein is a blood vessel, but we may 
assume that a vein is a blood vessel, and it can be stated thus: 

B= BA 



192 DISJUNCTIVES. [Chap. 13. 

Now, if B = BA, then the combination aBC is inconsistent 
because it implies that B = a, and we therefore eliminate it 
by making a figure 7 in that section. 

Again, if B = A, then the combination aBc is inconsistent 
because it implies that B = a, and we therefore eliminate it by 
making a figure 8 in that section. 

Our definition of B now is, 

B = BAc, which translated is: 

A vein is a blood vessel and not an artery. 

The combination abc remains automatically. 

From it we can get the following definitions: 

(1) a = abc, which translated is, 

What is a not-blood vessel is neither a vein nor an artery. 

(2) be = abc, which translated is: 

Whatever is neither a vein nor an artery is not a blood 
vessel. 

This last definition is true if in the original proposition given 
us the alternates were all stated. We have worked out this 
example for the purpose of showing that it is not necessary to 
state, in the first instance, all the propositions which may be 
worked in the Reasoning Frame; we can pursue our course and 
as we perceive new propositions to be true, we can state them 
and work them out in the Frame. 

We saw in working out the proposition "Every blood vessel 
is either a vein or an artery'' that we obtained the proposition 
"A blood vessel which is not a vein is an artery." We can read 
this, "If any blood vessel is not a vein, then it is an artery." 

369. From a disjunctive proposition we can usually get 
equivalent hypothetical propositions. But a writer says, "Dis- 
junctive judgments cannot be reduced to hypotheticals." In 
this I think he is mistaken. The two propositions, "A blood 
vessel which is not a vein is an artery," "If a blood vessel is 
not a vein it is an artery," are equivalents. 

I have no doubt that disjunctives can usually, and perhaps 
always, be reduced to hypotheticals and both to categoricals. 

Let us make an ABC diagram, and eliminate the combina- 
tions ABC and Abc by making an X in those sections: 



§ 370.] 



EQUIVALENTS. 



193 



AB 


Ab 


aB 


ab 




X 








C 




X 






c 



Fig. 116. 



There are now two A-combinations, that is, combinations 
containing A, in the Seasoning Frame. They are ABc, AbC. 
Now we can read them categorically: 

A which is B = c 
A which is b = C 
Disjunctively we can read them, 

A = either B and c | C and b 
Hypothetically we can read them, 

If A = B it = c 
If A ±= b it = O 
This tends to prove that the three terms categorical, dis- 
junctive and hypothetical, refer to three different ways of read- 
ing the same propositions. 

370. A writer says, "It has, however, already been pointed 
out that two negative propositions do not in any sense state 
an alternative, since the denial of an alternative is equivalent 
to the affirmation of a conjunction; hence, if a complex alterna- 
tive proposition is defined as a proposition which states an 
alternative by means of an alternative predicate, then only 
affirmative propositions can fall into this category. Distinc- 
tions of quantity cannot be applied to true compound alterna- 
tives, nor can distinctions of quality.'' 

If I understand the meaning of this statement, which I prob- 
ably do not, the writer would deny the validity of propositions 
like these: 

A | b = C | d, and conversely, 
13 



194 



DISJUNCTIVES. 



a | b = c | d, and conversely. 
If so, I think he is mistaken. 
Let us make an ABCD diagram 



AB 


Ab 


aB 


ab 






2 


2 




CD 


1 






1 


Cd 


1 






1 


cD 




2 


2 




cd 



Fig. 117. 



[Chap. 13. 



A | b = C | d, means AB | ab = CD | cd, because A | b 
means A without b, and that is the same as saying A with 
B, on the principle that two negatives make an affirmative. 
Without b is equivalent to B in this case. It is a general prin- 
ciple of logic that two negatives are equal to an affirmative; 
not not-A means A. 

Now we would express A without B thus: Ab; and A or B 
means A without B or B without A, that is, Ab or aB. 

Now, if A without B means Ab, then A without b must mean 
AB. AB means A without b. 

Again, A or b means, b without A, that is, ab, so that the- full 
expression of A or b is, 
AB or ab 

Again, the full expression of C or d is, 
CD | cd 

I assume that we can state the proposition, 
A | b = C | d, conversely, thus: 

(1) AB | ab = CD | cd 

(2) CD | cd = AB | ab 

Now, if AB | ab = CD | cd, then the combinations ABCd, 
ABcD, abCd, abcD are inconsistent because they say that AB 
| ab = Cd | cD, and we therefore eliminate them by making 
a figure 1 in those sections. 



§ 371.J 



NEGATIVE DISJUNCTIVES. 



195 



Again, if CD | cd = AB | ab, then the combinations AbCD, 
aBCD, Abed, aBcd are inconsistent, because they mean that 
CD | cd = Ab | aB, and we therefore eliminate them bv mak- 
ing a figure 2 in those sections. 

From the combinations which remain in the Reasoning 
Frame we can get the following definitions: 

(1) Ab = Cd | cD 

(2) aB = Cd | cD 

(3) Cd = Ab | aB 

(4) cD = Ab | aB 

(5) AB = CD | cd 

(6) ab = CD | cd 

(7) CD = AB | ab 

(8) cd = AB | ab 

As all the letters remain in the Reasoning Frame our pre- 
mises are consistent, and therefore the propositions are valid 
propositions. 

371. Our second proposition was, 

a or b is c or d, and conversely, which can be stated, 
thus: 

(1) aB | Ab = cD | Cd 

(2) cD | Cd = aB | Ab 

Now, if aB | Ab === cD | Cd, then the combinations AbCD, 
aBCD, Abed, aBcd, are inconsistent because they imply that 
Ab | aB = CD | cd, and we therefore eliminate them from 
an ABCD diagram, by making a figure 1 in those sections: 



AB 


Ab 


aB 


ab 






1 


I 


CD 


2 






2 
2 


Cd 


2 




cD 




1 


1 




cd 



Fig. 118. 



196 



DISJUNCTIVES. 



[ Chap. 13. 



Again, if Cd | cD = Ab | aB, then the combinations 
ABCd, ABcD, abCd, abcD, are inconsistent because they imply 
that Cd | cD = AB | ab, and we therefore eliminate them by 
making a figure 2 in those sections. 

We have now eliminated the same combinations which the 
propositions, 

A | b =C | d, and 
C | d = A | b 

caused us to eliminate; the definitions which can be obtained 
from the combinations which remain will be the same, and this 
proves that the proposition a | b = c | d, is equivalent to the 
proposition A | b = C | d. 

372. Let us take this proposition: 
a or b is c or I), and conversely, 
We can state it thus: 

(1) Ab | aB = cd | CD 

(2) cd ) CD = Ab | aB 

Now, if Ab | aB = cd | CD, then the combinations AbCd, 
AbcD, aBCd, aBcD are inconsistent, because they imply that 
Ab | aB = Cd | cD, and we therefore eliminate them from 
an ABCD diagram by making a figure 1 in those sections. 



AB 
o 


Ab 


aB 


ab 




1 




o 


CD 




1 
1 




Cd 




1 




cD 


2 






2 


cd 



Fig. 119. 



Again, if cd | CD = Ab | aB,then the combinations ABCD 
ABcd, abCD, abed, are inconsistent because they imply that 
cd | CD = AB | ab, and we therefore eliminate them by 
making a figure 2 in those sections. 



§ 373.J 



NEGATIVE DISJUNCTIVES. 



197 



From the combinations which remain we can get the follow- 
ing definitions: 

(1) Ab == AbCD | Abed 

(2) aB = aBCD | aBcd 

(3) cd = cdAb | cdaB 

(4) CD = CDAb | CDaB 

(5) AB = ABCd | ABcD 

(6) ab == abCd | abcD 

(7) Cd == ABCd | abCd 

(8) cD = ABcD | abcD 

373. Let us take this proposition, 

a | B = O | D, and conversely, which can be stated thus: 

(1) ab | AB = Cd | cD 

(2) Cd | cD = ab | AB 

Make an ABCD diagram: 



AB 


Ab 


aB 


ab 




1 






1 


CD 




2 


2 




Cd 




2 


2 




cD 


1 






1 


cd 



Fig. 120. 



Now, if ab | AB = Cd | cD, then the combinations abCD, 
abed, ABCD, ABcd are inconsistent, because thpy imply that 
ab | AB = CD | cd, and we therefore eliminate them by 
making a figure 1 in those sections. 

Again, if Cd | cD = ab | AB, then the combinations AbCd, 
AbcD, aBCd, aBcD are inconsistent, because they imply that 
Cd | cD = Ab | aB and we therefore eliminate them by mak- 
ing a figure 2 in those sections. 

We have now eliminated the same combinations which 
the proposition a | b = c | D, caused us to eliminate; the def- 



198 DISJUNCTIVES. [ Chap. 13. 

inations are the same in both cases, and the propositions are 
therefore equivalent. 

This way of elucidating these disjunctive propositions is, I 
think, entirely new. 

374. We stated the definitions which we obtained in the last 
case, in the alternative form, but they can be reduced to cate- 
goricals. Whenever any letter appears in one combination 
only, it can be defined categorically by stating that it is the 
other letters in the combination. Thus, if we have the combi- 
nation Abe, and that is the only combination in which c 
appears, the definition of c is, cAb. Similarly, when we have 
any combination and that combination of letters appears only 
once, we can define it by the remaining letter or letters. Thus, 
if we have the combination ABCD and the group ABC appears 
only in that combination, then we can say that ABC is ABCD. 

If we have the following combinations: 

(1) ABCD 

(2) ABcd 

From these two combinations we can get the following cate- 
gorical definitions: 

(1) ABC = ABCD 

(2) ABc = ABcd 

or we can obtain the following disjunctive definition: 
AB = ABCD | ABcd 
Categoricals containing common terms can always be com- 
bined into disjunctives, and disjunctives can generally be 
expanded into categoricals. 

375. Let us take this example of a disjunctive proposition: 

A | B | C = D, and conversely. 
Now, this means A without B and without C, or B without A 
and without C, or C without A and without B, is D. 
It can be stated thus: 

(1) Abe | aBc | abC = D 

(2) D = Abe | aBc | abC 

Make an ABCD diagram: 



§ 375.] 



AN EXAMPLE. 



199 



AB 


Ab 


aB 


ab 




2 


3 


4 


CD 








1 


Cd 


5 






6 


cD 
cd 




1 


1 





Fig. 121. 



Now, if Abe | aBc | abC = D, then the combinations Abed, 
aBcd, abCd, are inconsistent, because they imply that the com- 
binations Abe, aBc, abC = d, and we therefore eliminate them 
by making a figure 1 in those sections. 

Again, if D = Abe | aBc | abC, then the combination 
ABCD is inconsistent, because it implies that D = ABC, and 
we therefore eliminate it by making a figure 2 in that section. 

Again, the combination AbCD is inconsistent, because it 
implies that D = AbC, and we therefore eliminate it by mak- 
ing a figure 3 in that section. 

Again, the combination aBCD is inconsistent, because it 
implies that D = aBC, and we therefore eliminate it by mak- 
ing a figure 4 in that section. 

Again, the combination ABcD is inconsistent, because it 
implies that D = ABc, and we therefore eliminate it by mak- 
ing a figure 5 in that section. 

Again, the combination abcD is inconsistent, because it 
implies that D = abc, and we therefore eliminate it by making 
a figure 6 in that section. 

We have now eliminated all the inconsistent combinations. 
From the combinations which remain we can get the following 

definitions: 

(1) abC | Abc | aBc == D 

(2) ABC = ABCd 

(3) ABc = ABcd 

(4) AbC = AbCd 



200 DISJUNCTIVES. [Chap. 13. 

(5) Abe = AbcD 

(6) abC = abCD, eto. 

376. Let us take this disjunctive proposition : 

a | b | G = D and conversely, which can be stated 

thus: 

(1) aBc | Abe | ABC == D 

(2) D = aBc | Abe | ABC 

The reader will understand from previous explanations, that 
in this case a means without b and without C; and that without 
b means with B; and without C means with c, so that the full 
logical expressionof a is aBc. 

Again, b means without a and without C; without a means 
with A; without C means with c, so that the full logical expres- 
sion of b is Abe. 

Again, C means without a and without b, and that means, 
with A and with B, so that the full logical expression of C is 
ABC. 

Make an ABCD diagram: 



AB 


Ab 
2 


aB 


ab 






2 


2 


CD 


1 
2 






Cd 






2 


cD 




1 


1 




cd 



Fig. 122. 

Now, if aBc | Abe | ABC = D, then the combinations 
aBcd, Abed, ABCd, are inconsistent, because they imply that 
Abe | aBc | ABC = d, and we therefore eliminate them by 
making a figure 1 in those sections. 

Again, if D =*= aBc | Abe | ABC, then all the other combi- 
nations which contain D are inconsistent, and we eliminate 
them by making a figure 2 in those sections. 



§ 377.] 



AN EXAMPLE. 



201 



They are AbOD, aBCD, abCD, ABcD, abcD. 
From the combinations which remain we can get the follow- 
ing definitions : 

(1) D = ABC | Abe | aBc 

(2) ABC = ABCD 

(3) ABc = ABcD 

(4) AbC = AbCd, etc. 

377. Again, let us take this example: 

a | b | c = D | E and conversely, which can be stated 
thus: 

(1) aBC | AbC | ABc = De | dE 

(2) De | dE = aBC | AbC | ABc 

Make an ABCDE diagram: 



AB 


Ab 


aB 


ab 


CDE 




1 


1 




2 






2 


CDe 


2 






2 


CdE 




1 


1 




Cde 


1 








cDE 




2 


2 


2 


cDe 




2 


2 


2 


cdE 

cde 


1 









Fig. 123. 



Now, if aBC | AbC | ABc = De | dE, then the combina- 
tions aBCDE, aBCde, AbCde, AbCDE, ABcDE, ABcde, are 
inconsistent, because they imply that aBC | AbC | ABc 
= DE | de, and we therefore eliminate them by making a 
figure 1 in those sections. 



202 



DISJUNCTIVES. 



[Chap. 13. 



Again, if De | dE = aBC | AbC | ABc, then all the other 
combinations which contain De | dE are inconsistent. They 
are ABCDe, ABCdE, AbcDe, AbcdE, aBcDe, aBcdE, abCDe, 
abCdE, abcDe, abcdE, and we therefore eliminate them by 
making a figure 2 in those sections. 

From the combinations which remain we can get the follow- 
ing definitions: 

(1) De = AbC | aBC | ABo 

(2) dE = AbC | aBC | ABc 

( 3) aBC = De | dE 

(4) AbC = De | dE 

(5) Abe = DE | de 

(6) ABCD = ABCDE 

(7) ABCd = ABCde 
(S) ABcD = ABcDe 

(9) ABcd = ABcdE, etc. 
378. Let us now take still more complicated forms of Dis- 
junctive propositions: 

A = B | C = D 
= D | A = B 
I interpret this to mean A = B, except where C = D, 
A = B unless C = D, and C = D except where A = B, 
or C = D unless A = B. 

There are four combinations of CD, viz: CD, Cd, cD, cd. 
Therefore, the proposition A = B, except where C = D, 
means A = B where C = d, | c = D, | c = d. And the 
proposition C = D except where A = B means, C = D 
where A = b, | a =B, | a = b. 
Make an ABCD diagram: 



AB 


Ab 


aB 


ab 




13 


3 

2 






CD 




3 


3 


Cd 




2 


— 




cD 




2 




cd 



Fig. 124. 



§ 378.] A NEW METHOD. o 03 

The proposition A = B | C = D, also means chat the com- 
bination ABCD is inconsistent, because it means A = 1> and 
C = D, and we therefore eliminate it by making a figure 1 
in that section. 

Now, if A == B, except where C = D, then the combina- 
tions AbCd, AbcD, Abed are inconsistent, because they imply 
that A == b, except where C = D, and we therefore elimi 
nate them by making a figure 2 in those sections. 

Again, if C = D, except where A = B. then the combina- 
tions ABCD, AbCd, aBCd, abCd are inconsistent, because they 
imply that = d, except where A = B, and we therefore 
eliminate them by making a figure 3 in those sections. 

From the combinations which remain we can get the fol- 
lowing definitions: 

(1) AB = ABCd | ABcD | ABcd 

(2) CD == AbCD | aBCD | abCD 
which we can read: 

A = B | C = D, 
that is, A = B in every case except where C = D, and C = 
D in every case except where A = B. 

This form of a disjunctive proposition gave the old logici- 
cians considerable trouble. 

I think this method of solving it is quite new. 



CHAPTER XIV. 

OR. 

379. The little word "or" has given as much trouble to 
logicians as any other word in the English language. Before 
proceeding to discuss its meaning, I thought best to give a 
number of practical examples to show our method of working 
Disjunctive propositions. 

Many logicians have contended that the usual meaning of 
"or" was "one or the other or both." Others have insisted that 
it meant "one or the other and not both." 

380. Prof. Hamilton said that all disjunctives are to be re- 
garded as exclusive, that is, when we say, 'All A is X or Y,' 
we are not only justified in inferring that any A which is not 
X is Y, but also that any A which is X is not Y." 

381. Prof. Venn says, Symbolic Logic, p. 48, "Thus to say, 
'He is deceiver or deceived' is by no means the same thing as 
to say, 'He is deceiver and deceived.' " 

Prof. Venn also says that "or" means "one or the other or 
both." 

382. When we take the position that "or" means one or the 
other and not both, except in those cases where it is expressly 
stated that "or" means "one or the other or both," it is easy to 
be consistent in our interpretation of the word "or" wherever 
it occurs. But if we take the other interpretation, i. e., that it 
means one or the other or both, we shall find it difficult to 
maintain our consistency, and shall oscillate from one meaning 
to the other. 

383. If a man should say, "Arches are circular or pointed 
or both," we would naturally conclude that he did not know 
well what he was talking about, and I think that it is generally 



§ 384-389.] EXCLUSIVENESS. o 5 

other or both" he does so because he is more or less ignorant 
of his subject. 

384. St. Aquinas held to the opinion that when "or" did not 
mean "one or the other and not both" that the proposition was 
false, and Kant held the same opinion. This meaning of "or' 1 
is called the "exclusive meaning; the other is called the -non 
exclusive." 

385. Prof. Boole also took the "exclusive" side, but Arch- 
bishop Whately, Prof. Mansel and J. S. Mill have taken the 
"non-exclusive" side. 

Archbishop Whately gives this example: 
"Virtue tends to procure us either the esteem of mankind or 
the favor of God." 

386. Prof. Jevons says in "Principles of Science," p. 68, "1 
discuss this subject fully, because it is really the point which 
separates my logical system from that of Boole." 

387. When a man thinks that "or" means "one or the other 
or both," when he comes to state an alternative proposition he 
ought to say of his subject that it is "one or the other or both." 
If he says, "Matter is solid or liquid" and means that it is 
"solid or liquid or both," then he ought to say expressly, that 
"matter is solid or liquid or both." If he says "one or the 
other or both" we can state that expression symbolically and 
solve our logical problems just as correctly and easily as when 
"or" is used in the exclusive sense. 

388. There is another meaning to the word "exclusive" for 
if we say, "one or the other or both," then "or" means that only 
one of those alternatives can be true; whichever alternative is 
the true meaning, it excludes both the others. 

389. Miss Jones, a very talented logician, says in "Elements 
of Logic," p. 117, that besides Kant and Hamilton, Thompson, 
Bain and Fowler take the exclusive view. She gives this ex- 
ample and says: 

" 'Every ragged person either is poor or wishes to be 
thought poor.' This seems to me an extremely ingenious « \ 



•206 OR. [ Chap. 14. 

ample, and at first sight very telling on the side of unexclusive- 
ness; but if its full meaning were expressed, would it not run 
as follows: 'Every ragged person is ragged either because he 
is poor, or because he wishes to be thought poor,' and in this 
case the alternation is exclusive." 

She says, "Alternatives must always have some element of 
exclusiveness, otherwise they have no logical value whatever." 

Prof. Keynes takes the unexclusive side. 

390. In our system when we wish to state that A = B | C 
but not both, we do it in this way : 

A == Be | bC 
And when we wish to state that A = B | C or both, we can 
do it in either of the following ways: 

(1) A = Be | bC | BG 

(2) A ■-= B | bC 

In the first case the combinations ABC and Abe are incon- 
sistent; in (1) and (2) the combination Abe is the only incon- 
sistent combination. 

391. I think that "or" is indefinite and that sometimes it is 
used exclusively and sometimes unexclusively. Unless the 
unexclusive moaning is plainly indicated I always give it the 
exclusive meaning. 

EXERCISES. 

392. (1) What is a complete equivalent for a | b = c | d? 

(2) What is a complete equivalent for AB =CD? 

(3) What is a complete equivalent for a | B = c | D? 

(4) What are the categorical equivalents for the following 
alternative proposition : 

a = aBCd | aBcd | abCD | abcD? 

(5) What are the categorical equivalents for the following 
alternative proposition : 

B = ABCD | ABcd | aBCd | aBcD? 

(6) What alternative proposition is an equivalent for the fol- 
lowing categoricals: 

ABCD, AbCD, aBCD? 



§ 392.] EXERCISES. 207 

(7) What alternative proposition is equivalent to the fol- 
lowing categoricals : 

Acdb, dcba, bade? 

(8) What conclusions can be drawn from the following pair 
of propositions: 

G = B | A 
B == BA? 

(9) W'hat conclusions can be drawn from the following pair 
of propositions : 

B = either A | 
B = a? 

(10) What conclusions can be drawn from the following pair 
of propositions : 

A | D = B | E 
C | F = A | D? 

(11) What conclusions can be drawn from the following pair 
of propositions: 

a | b = a | o 
a = C? 

(12) What conclusions can be drawn from the following pair 
of propositions : 

A | B | C = d 
d == A | B | C? 

(13) What conclusions can be drawn from the following pair 
of propositions : 

A | B = CD | cd 
C I D = AB I ab? 



CHAPTER XV. 

HYPOTHETICAL PROPOSITIONS, 

393. Hypothetical propositions commonly indicate a certain 
amount of doubt and usually commence with the word "if," 
"when," "where," "whenever," "wherever," "given," "granted," 
"provided," "since," or some other similar conjunction. 

394. "If A = B" is an indefinite statement. It denies noth- 
ing and has little if any more predicative force than an identi- 
cal proposition such as AB = AB. It does not mean that A 
= B. 

395. A hypothetical proposition can be converted into a cate- 
gorical proposition by the use of the words "the case of." 

If Caesar was an usurper he deserved death, is a hypothetical 
proposition. It can be converted into the categorical propo- 
sition, the case of Caesar being an usurper is a case of 
Caesar deserving death. 

396. A hypothetical proposition usually has a suppressed 
premise and when this is supplied we can convert the hypo- 
thetical proposition into two categorical propositions and then 
deduce the conclusion in the usual way. In the hypothetical 
proposition, 

If Caesar was an usurper he deserved death, the implied pre- 
mise is, 
All usurpers deserve death. 

397. Let us take this example: 

(1) The case of Caesar is the case of an usurper, 

(2) All usurpers deserve death, therefore Caesar deserved 
death. 

Let A = the case of Caesar, 

B = the case of an usurper, 
C = deserved death. 



§ 398.] 



AN EXAMPLE. 



209 



The propositions can be stated thus: 

(1) A = AB 

(2) B =-. BO 
Make an ABC diagram: 



AB 


Ab 


aB 


ab 






1 






C 


2 


1 


2 




c 



Fig. 125. 

Now, if A = AB, then the combinations containing Ab are 
inconsistent, and we eliminate them by making a figure 1 in 
those sections. 

Again, if B = BC, then the combinations containing Be are 
inconsistent and we eliminate them by making a figure 2 in 
those sections. 

From the uneliminated combinations we can get these defi- 
nitions : 

(1) AB = ABC, which can be translated 

The case of Caesar being the case of an usurper, is a case 
where Caesar deserved death. 

(2) c = cb, which can be translated: 

One not deserving of death is not the case ol an usurper. 

398. Let us take this example: 

(1) Caesar was an usurper, 

(2) All usurpers deserve death, therefore Casar deserved 
death. 

Let A = usurper, 

B = deserved death, 

C = Caesar. 
The propositions can be stated thus: 

(1) C = CA 

(2) A = AB 
14 



210 HYPOTHETICAL PROPOSITIONS. 

Make an ABO diagram: 



[ Chap, 15. 



AB 


Ab 


aB 


ab 






2 


1 


1 


C 




2 






c 



Fig. 126. 

Now, if = CA, then the combinations containing Ca are 
inconsistent and we eliminate them by making a figure 1 in 
those sections. 

Again, if A = AB, then the combinations containing Ab are 
inconsistent and we eliminate them by making a figure 2 in 
those sections. 

From the uneliminated combinations we can get this defi- 
nition, 

C = CB, which can be translated: 
Therefore, Caesar deserved death. 

399. Let us take this example, 

(1) If Caesar was an usurper he deserved death, 

(2) Caesar was an usurper. 
Let A = Caesar, 

B = usurper, 
C = deserved death. 
The propositions can be stated thus : 

(1) AB = ABC, 

(2) A = AB 

Make an ABC diagram : 



§§400,401.] 



CONDITIONALS. 



211 



AB 


Ab 


aB 


ab 






2 






C 


1 


2 






c 



Fig. 127. 



Now, if AB = ABC, then the combination ABc is incon- 
sistent and we eliminate it by making a figure 1 in that section. 

Again, if A = AB, then the combinations containing Ab 
are inconsistent, and we eliminate them by making a figure 2 
in those sections. 

From the uneliminated combinations we can get this defini- 
tion of A, 

A = C, which can be translated: 
Therefore, Csesar deserved death. 

Whenever we have a symbolic proposition reading, If A = 
B, A = C, we can always state it in this categorical form, 
AB = ABC 

This means where we have AB we must have ABC. 

400. Many logicians divide hypothetical propositions into 
hypothetical and conditionals. 

A conditional proposition relates to some prior circumstance, 
for example : 

If a child is spoiled his parents suffer. 
The hypothetical implies a relation which always exists, for 
example : 

"If patience is a virtue, there are painful virtues," but 
in our system we treat them both alike. 

401. Let us take this example: 

(1) If A = b then A = c 

(2) A = C 
The propositions can be stated thus: 

(1) Ab = Abe 

(2) A = AC 



212 



HYPOTHETICAL PROPOSITIONS. 



[ Chap. 15. 



Make an ABC diagram: 



AB 


Ab 


aB 


ab 






1 






C 


2 


2 






c 



Fig. 128. 

Now, if Ab = Abe, then the combination AbC is inconsist- 
ent and we eliminate it by making a figure 1 in that section. 

Again, if A = AC, then the combinations containing Ac are 
inconsistent and we eliminate them by making a figure 2 in 
those sections. 

From the uneliminated combinations we can get this defi- 
nition of A, " 

A = AB 

402. Let us take this example : 

(1) if A = B then A = C 

(2) A = b 

The propositions can be stated thus: 
(L) AB = ABC 
(2) A = Ab 
In this case the minor premise denies the antecedent 
of the major premise. The rule is, that from the denial of the 
antecedent the truth of the consequent cannot be inferred. 
Make an ABC diagram: 



AB 


Ab 


aB 


ab 




2 









C 
c 


2 

1 







Fig. 129. 



§ 402.] 



DENIAL OF THE CONSEQUENT. 



213 



Now, if AB = ABC, then the combination ABc is inconsist- 
ent and we eliminate it by making a figure 1 in that section. 

Again, if A = Ab then the combinations containing AB are 
inconsistent and we eliminate them by making a figure 2 in 
those sections. 

From the uneliminated combinations we can get this defi- 
tion of A : 

A = G | o 

403. Let us take this example: 

(1) If A = B then A = 

(2) A = c 
Therefore, A = b 

The propositions can be stated thus : 

(1) AB = ABO 

(2) A = Ac 

In this case the minor premise denies the consequent of 
major premise. The rule is that from the denial of the conse- 
quent, the contradictory of the antecedent can be inferred. 

Make an ABO diagram : 



AB 


Ab 


aB 


ab 




2 


2 






C 


1 








c 



Fig. 130. 



Now, if AB = ABC, then the combination ABc is inconsist- 
ent and we eliminate it by making a figure 1 in that section. 

Again, if A = Ac, then the combinations containing AC 
are inconsistent and we eliminate them by making a figure 
2 in those sections. 

From the uneliminated combinations we can get this defini- 
tion of A, 



214 



HYPOTHETICAL PROPOSITIONS. 



[ Chap. 15. 



A = Ab 

404. Let us take the following example: 

(1) If A = b then A = c 

(2) A = b, therefore, 

A = c 

The propositions can be stated thus: 

(1) Ab = Abo 

(2) A == Ab 

Make an ABC diagram : 



AB 


Ab 


aB 


ab 




2 


1 






C 


2 








c 



Fig. 131. 



Now, if Ab = Abe, then the combination AbO is incon- 
sistent and we eliminate it by making a figure 1 in that sec- 
tion. 

Again, if A = Ab, then the combinations containing AB 
are inconsistent and we eliminate them by making a figure 2 
in those sections. 

From the uu eliminated combinations we can get this defi- 
nition of A: 

A = Ac 

The propositions in this case are equivalent to the propo- 
sitions in the last preceding case, because they eliminate the 
same combinations. 

405. Let us take this example.- 

(1) If A = B then A == 

(2) A = C 
The propositions can be stated thus: 



§ 406.] 



AN EXAMPLE. 



215 



(1) AB = ABC 

(2) A = AC 

In this case the minor premise affirms the consequent of the 
major premise. The rule is, that from the affirmation of the 
consequent, the truth of the antecedent cannot be inferred. 

Make an ABC diagram: 



AB 


Ab 


aB 


ab 












C 


1 


2 






c 


2 











Fig 132. 



Now, if AB = ABC, then the combination ABc is inconsist- 
ent and we eliminate it by making a figure 1 in that section. 

Again, if A = AC, then the combinations containing Ac 
are inconsistent and we eliminate them by making a figure 2 in 
those sections. 

From the uneliminated combinations we can get this defi- 
nition of A, 

A = B | b. 

406. Let us take this example from Archbishop Whately, 

(1) If the first preachers of the gospel had displayed no 

miracles they could not have obtained a hearing. 

(2) But they did obtain a hearing. 

(1) Let A = first preachers of the gospel, 

(2) B = displayed miracles, 

(3) C = obtained a hearing. 

The propositions may be stated thus, 

(1) Ab = Abe 

(2) A = AC 

Make an ABC diagram: 



216 



HYPOTHETICAL PROPOSITIONS. [ Chap. 15. 



AB 


Ab 


aB 


ab 






1 






C 


2 


2 






c 



Fig. 133. 

Now, if Ab = Abo then the combination AbO is inconsistent 
and we eliminate it by making a figure 1 in that section. 

Again, if A = AG, then the combinations containing Ac are 
inconsistent and we eliminate them by making a figure 2 in 
those sections. 

From the uneliminated combinations we can get this defi- 
nition of A, 

A = AB, which can be translated: 

Therefore, the first preachers of the gospel displayed mira- 
cles. 

407. Let us take this example from Prof. Bain : 

(1) If the education of certain children is neglected, then 
they will grow up ignorant, 

(2) The education of certain children has been neglected. 

(1) Let A = education of certain children, 

(2) B = neglected, 

(3) = ignorant. 

The propositions can be stated thus: 

(1) AB == ABO 

(2) A = AB 



Make an ABO diagram: 



408.] 



AN EXAMPLE. 



217 



AB 


Ab 


aB 


ab 




1 


2 






C 


2 






c 



Fig. 134. 

Now, if AB = ABO, then the combination ABc is inconsist- 
ent and we eliminate it by making a figure 1 in that section. 

Again, if A = AB, then the combinations containing Ab 
are inconsistent and we eliminate them by making a figure 2 in 
those sections. 

From the uneliminated combinations we can get this defi-. 
nition of A, 

A = ABC, which can be translated: 

The education of certain children has been neglected and 
they will grow up ignorant. 

408. Let us take this example from Prof. Bain: 

(1) If the weather continues fine we shall go to the country. 

(2) The weather continues fine. 

Let A = weather, 

B = continues fine, 

C == we, 

D = shall go to the country. 

The proposition means, where we have AB we must have 
ABCD, and it may be stated thus: 

(1) AB = ABCD 

(2) A == AB 



Make an ABCD diagram: 



218 



HYPOTHETICAL PROPOSITIONS. 



[ Chap. 15. 



AB 


Ab 


aB 


ab 






2 




CD 


1 


2 






Cd 


1 


2 






cD 


1 


2 






cd 



Fig. 135. 

Now, if AB = ABCD, then the combinations containing 
ABCd, ABc, are inconsistent and we eliminate them by making 
a figure 1 in those sections. 

Again, if A = AB, then the combinations containing Ab 
are inconsistent and we eliminate them by making a figure 2 
in those sections. 

From the uneliminated combinations we can get this defi- 
nition : 

AB = ABCD, which can be translated: 

The weather continues fine, therefore, we are going into the 
country. 

409. In hypothetical propositions the granting of the conse- 
quent does not prove the truth of the antecedent, for example : 

"If he caught the infection he will die." 

His death does not prove that he caught the infection, be- 
cause there are many causes of death besides the one men- 
tioned. 

Let A = he, 

B = caught the infection, 
C = will die 
We can state the premises thus : 

(1) AB == ABO 

(2) A = AG 

Make an ABC diagram : 



§ 410.] 



AN EXAMPLE. 



219 



AB 


Ab 


aB 


ab 












C 


2 

1 


2 






c 



Fig. 136. 

Now, if AB = ABC, then the combination ABo is inconsist- 
ent and we eliminate it by making a figure 1 in that section. 

Again, if A = AC, then the combinations containing Ac 
are inconsistent and we eliminate them by making a figure 2 
in those sections. 

From the uneliminated combinations we can get this defi- 
nition of A, 

A = B | b 

The result proves that the granting of the consequent has 
not established the truth of the antecedent, viz: He caught the 
infection. 

410. Let us take this example: 

(1) If force is expended an equivalent force will be gene- 
rated. 

(2) An equivalent force is generated. 

(1) Let A = force, 

B = expended. 

C = equivalent force, 

D = generated. 

The premises can be stated thus: 

(1) AB = ABCD 

(2) C = CD 

Make an ABCD diagram: 



220 



HYPOTHETICAL PROPOSITIONS. 



[ Chap. IS. 



AB 


Ab 


aB 


ab 












CD 
Cd 


1 
2 


2 


2 


2 


1 








cD 


1 








cd 



Fig. 137. 



Now, if AB = ABCD, then the combinations containing 
ABCd, ABc, are inconsistent and we eliminate them by making 
a figure 1 in those sections. 

Again, if C = CD, then the combinations containing Cd are 
inconsistent and we eliminate them by making a figure 2 in 
those sections. » 

From the uneliminated combinations we can get this defi- 
nition of A, 

A = B | b, which can be translated: 

Force is expended or not expended. 

This furnishes another example of the rule that the affirm- 
ation of the consequent does not establish the truth of the 
antecedent. 

In this case we know that as a matter of fact when force is 
generated, force was expended; that is a matter of fact which 
science teaches but it is not a logical conclusion from the 
given premises. 

411. Let us take this example: 

(1) If this river has tides, the sea into which it flows 
must have tides. 

(2) This river has tides. 

(1) Let A = this river, 

(2) B = river tides, 

(3) C =±= the sea into which it flows, 

(4) D — sea tides. 
The premises can be stated thus: 



§412.] 



AN EXAMPLE. 

(1) AB = ABCD 

(2) A = AB 



221 



Make an ABCD diagram: 



AB 


Ab 


aB 


ab 






2 






CD 


1 


2 






Cd 


1 








cD 


1 






cd 



Fig. 138. 

Now, if AB = ABCD, then the combinations containing 
ABCd, ABc are inconsistent and we eliminate them by making 
a figure 1 in those sections. 

Again, if A = AB, then the combinations containing Ab 
are inconsistent and we eliminate them by making a figure 2 
in those sections. 

From the uneliminated combinations we can get this defi- 
nition of A, 

A == BCD, which can be translated, 

The river has tides and the sea into which it flows has 
tides. 

412. Let us take this example: 

(1) If this river has tides, the sea into which it flows 
must have tides. 

(2) The sea into which it flows has not tides. 

I think the proposition if A = B then C = D means that 
the only AB there is, is the one which is CD. 
The premises can be stated thus: 

(1) AB = ABCD 

(2) C = Cd 



Make an ABCD diagram: 



222 



HYPOTHETICAL PROPOSITIONS. 



[Chap. 15. 



AB 


Ab 


aB 


ab 




2 


2 


2 


2 


CD 


1 








Cd 


1 








cD 


1 








cd 



Fig. 139. 



Now, if AB =ABCD, then the combinations containing 
ABCd, ABc are inconsistent and we eliminate them by mak- 
ing a figure 1 in those sections. 

Again, if C = Cd, then the combinations containing CD are 
inconsistent and we eliminate them by making a figure 2 in 
those sections'. 

From the uneliminated combinations we can get this deli 
nition of A, 

A = b, which can be translated: 

This river has not tides. 

413. Hypothetical propositions are of no value except for 
the categorical information which they contain. 

It has been said that in the hypothetical proposition, 
If A = B, then it == C, 
conveys the categorical information that 

All B =± :0 

but I think this is a mistake. The categorical information 
conveyed is, that where we have the case of AB, we have also 
a case of C. 

414. Prof. Venn gives this good illustration: 
"Suppose some one had said in 1852, 'If Louis Napoleon 

becomes emperor he will be crowned'." 

While it is true as a matter of fact, that all emperors are 
crowned, this is not a logical inference from the proposition, 
because, as Prof. Venn well says, "If one were to say in 1S52 
'If Napoleon becomes emperor he will be perjured'," we could 
not logically say, all emperors are perjured. 



£§ 415-418.] 



UNIVERSALS. 



223 



415. Miss Jones says, in "Elements of Logic," p. 113, (I have 
changed the letters,) "What any conditional proposition, 

If any A is B, that A is C, 
seems to me to assert is, that the Cness of any A is an inference 
from its being AB, thus, any conditional would be universal 
and affirmative. I think also that it implies the existence 
of some A's are not B's." 

41 G. Unless there is something to indicate to the contrary, 
I treat all hypothetical propositions, and disjunctive propo- 
sitions, excepting disjunctive propositions having the form of 
A = B | G = D, as Universals. I do not, however, always 
repeat the subject on paper before the predicate, on account of 
the tediousness of the process. 

By this time the reader will have learned that in the case of 
Universals, the subject is to be repeated before the predicate, 
at least in the mind, whether it is put down on paper or not. 

417. I usually treat disjunctive propositions of the form, 

A = B | C = D, as being doubly universal or reciprocating 

propositions, i. e., they are to be read and worked both forward 

and backward, thus: 

A = B | C =D 

C = D|A-B 

418. There is no doubt that the proposition, 

If any A is B, that A is C, implies the existence of some 
A's which are not B's. 

Make an ABC diagram. 



AB 


Ab 


aB 


ab 












C 


1 








c 



Fig. 140. 



224 HYPOTHETICAL PROPOSITIONS. [Chap. 15. 

Now, if where A = B that A = C, then the combination 
APc is inconsistent, because it implies that where A = B 
A — c, and we eliminate it by making a figure 1 in that sec- 
tion. 

The Reasoning Frame now shows that in two cases A = b. 

419. In hypothetical propositions the "if" of the antece- 
dent may generally be replaced by "when" or "where" or 
"whenever" or "wherever" or "in the case in which," at our 
option. 

420. Dr. Keynes says that a hypothetical is composed of 
two propositions and that in the form, 

If A is true, then G is true, 
A and C stand for propositions and the words, "is true" are 
introduced to make this clear. 

421. In the old logic the quality of the consequent deter- 
mines the quality of the hypothetical proposition, 

IfA = BG = D 
is affirmative, 

If A = B = d 
is negative, 

422. We are liable to make the mistake that the two propo- 
sitions, 

If A = B = D 
If A = B C = d 
are contradictories. We have seen that the test of contra- 
dictoriness is the total elimination of a letter-term from the 
Reasoning Frame. 

But these two propositions do not cause the total elimina- 
tion of any letter-term. The one asserts that where A = £ 
C = D; the other asserts where A = B O = d. 
The inference which results is, that No A «= B. 

Make an ABCD diagram: 



§ 423.] 



CONVERSION. 






AB 


Ab 


aB 


ab 




2 




CD 


1 
2 

1 
2 










Cd 





cD 


1 
2 






cd 



Fig. 141. 

Now, if A = B C = D, then the combinations containing 
ABCd, ABc, are inconsistent, and we eliminate them by mak- 
ing a figure 1 in those sections. 

Again, if A = B C = d, then the combinations containing 
ABCD, ABc, are inconsistent, and we eliminate them by mak- 
ing a figure 2 in those sections. 

The result proves that A = b, No A = B. 

423. Hypothetical propositions can be converted. 
Let us take this example: 

If A = B then A = C 
It can be stated thus: 

AB = ABO 

Make an ABC diagram: 



AB 


Ab 


aB 


ab 












C 


1 








c 



Fig. 142. 



Now, if AB = ABC, then the combination ABc is inconsis- 
tent, and we eliminate it by making a figure 1 in that section. 
We can now read in the Reasoning Frame, 
If A == c, then A = b 
15 



226 



HYPOTHETICAL PROPOSITIONS. 



[Chap. 15. 



This is called the converse of the original proposition. 

424. Let us take this example from Dr. Keynes' "Formal 
Logic," p. 219: 

"If a straight line falling upon two other straight lines 
make the alternate angles equal to one another, these two 
straight lines shall be parallel." 

Let A = a straight line falling upon, 
B = two other straight lines 
C = make the alternate angles equal, 
D = parallel. 

We have already seen that the denial of the consequent is 
the denial of the antecedent, so that we can get this proposi- 
tion, 

If these two straight lines are not parallel, a straight line 
falling upon them will not make the alternate angles equal to 
one another. 

The premises can be stated thus: 

(1) ABC = ABCD 

(2) Bd == ABc 

Make an ABCD diagram : 



AB 


Ab 


aB 


ab 












CD 


1 
2 




2 




Cd 










cD 






2 




cd 



Fig. 143. 

Now, if ABC s= ABCD, then the combination ABCd is incon- 
sistent, and we eliminate it by making a figure 1 in that 
section. 

If Bd = ABc, then the combinations containing aBd, ABCd, 
are inconsistent, and we eliminate them by making a figure 
2 in those sections. 



§§ 425, 426.] 



EQUIVALENTS. 



227 



We can now get the following definitions in the Reasoning 
Frame: 

(1) ABO = ABCD 

(2) Bd = ABc 

(3) ABc = BD | Bd 
which can be translated, 

A straight line falling upon two other straight lines which 
does not make the alternate angles equal, then the two straight 
lines are parallel or not parallel. 

425. I believe that all the problems in geometry can be 
solved by our system, and that its use by mathematicians will 
bring to light a great many new geometrical truths, but, not 
being a mathematician myself, if I discovered a new truth in 
geometry, I would not know it. 

426. It has been said that either C is true or A is not true 
is a disjunctive equivalent for, if A is true, then C is true. 

The propositions can be stated thus: 

(1) If A = B, C = D 



(2) == D 
Make an ABCD diagram: 



A == b 



AB 


Ab 


aB 


ab 






2 






CD 


21 




2 


2 


Cd 


21 
21 






cD 








cd 



Fig. 144. 

Now, if where A = B G == D, then the combinations 
containing ABCd, ABc, are inconsistent, and we eliminate 
them by making a figure 1 in those sections. 

If C = D | A = b, and conversely, then the combinations 
containing ABCd, ABc, AbCD, aBCd, abCd, are inconsistent 
and we eliminate them by making a figure 2 in those sections. 



228 HYPOTHETICAL PROPOSITIONS. [Chap. 15. 

The Reasoning Frame now shows that the propositions are 
not equivalent because they do not eliminate the same combi- 
nations. 

The proposition C = D | A = b, is a much more definite 
proposition than the other, because it eliminates more combi- 
nations. But (1) is an inference from (2) because all the com- 
binations eliminated by (1) are eliminated by (2), but not con- 
versely. 

427. In "Formal Logic," p. 223, Dr. Keynes says: "Mr. 
McColl writes (the lettering is mine) the expression, If A then 
B may be read, A implies B, or if A is true B must be true. The 
statement If A then B, implies a or B. But it may be asked, 
are not the two statements really equal; ought we not therefore 
to write, If A then B = a | B? Now. if the two statements 
are really equivalent their denials will also be equivalent. 

Let us see if this will be the case, taking as concrete exam- 
ples: 

'If he persists in his extravagance he will be ruined;' 'He 
will either discontinue his extravagance or he will be ruined.' 

The denial of, If A then B is (the contradictory of If A then 
B), and this denial may be read, 'He may persist in his extrava- 
gance without necessarily being ruined.' 

The denial of a or B is Ab, which may be read, 'He will per- 
sist in his extravagance and he will not be ruined.' 

Now, it is quite evident that the second denial is a much 
stronger and more positive statement than the first. The first 
only asserts the possibility of the combination Ab ; the second 
asserts the certainty of the same combination. The denials of 
the statement, If A then B and a or B, having thus been proved 
to be not equivalent, it follows that the statements If A then B 
and a or B are themselves not equivalent, and that though a or 
B is a necessary consequence of, If A then B, yet, If A then B 
is not a necessary consequence of a or B (see Mind, 1880, pp. 
50-54; one or two slight verbal changes have been made in this 
quotation)." 

Let us now take up this statement, "If A then B implies a or 



§ 428.] 



EQUIVALENTS. 






B." The A in this case is understood to stand for a propo- 
sition. Let it stand for the proposition "A is B." Lei the D 

in this case stand for the proposition "C is D." 
We can now state, If A then B, thus : 

(1) If A = B then C = D 
and we can express the statement a or B, thus: 
(2) A = b | C = D 
The propositions can be stated thus: 

(1) AB = ABCD 

(2) A = b | C = D 

Make an ABCD diagram: 



AB 


Ab 


aB 


ab 






2 






CD 


21 




2 


2 


Cd 


21 








cD 


21 








cd 



Fig. 145. 

Now, if AB = CD, then the combinations containing ABCd, 
ABc, are inconsistent, and we eliminate them by making a ii^ r 
ure 1 in those sections. 

Again, if A = b, except where C = D, and conversely, then 
the combinations containing ABCd, ABc, AbCD, aCd, are 
inconsistent, and we eliminate them by making a figure 2 in 
those sections. 

The Seasoning Frame now shows that the two propositions 
are not equivalent because they do not eliminate the same 
combinations. But (1) is an inference from (2), 

428. Dr. Keynes says in the note on p. 223, of "Formal 
Logic," "Mr. Welton accepts Miss Jones' view up to a certain 
point, but apparently does not recognize all that it Invokes 
and hence obtains inconsistent results. 



230 HYPOTHETICAL PROPOSITIONS. [ Chap. 15. 

He regards, 

(1) A or B, i. e., A is B or C is D, 

(2) If not-A then B, i. e., if A = b C = D, as equivalents. 
For the contradictory of (1) he gives, 

(3) Neither A nor B, i. e., Neither A = B nor C = D, 
and he considers that (2) yields as its contradictory, 

(4) If not-A then not-B, i. e., If A = b C =d. 
This again being equivalent in his view to 

(5) A or not-B, i. e., A = B or C = d." 

The explanations following the propositions are mine. 

Dr. Keynes further says, "But (3) and (5) are obviously not 
equivalents. We may take Mr. Welton's concrete example 
(Logic, p. 281). 

The propositions, 

(a) This pen is either cross-nibbed or corroded by the ink, 
and, 

(b) This pen is neither cross-nibbed nor corroded by the ink, 
are given as contradictories, but (a) is regarded as equivalent 
to, 

(c) If this pen is not cross-nibbed, it is corroded by the ink. 
And for the contradictory of (c) Mr. Welton would give, 

(d) If this pen is not cross-nibbed, it is not corroded by the- 
ink. 

But (b) and (d) are clearly not equivalent to one another." 

Let us work out these examples, 
(1) cau be stated thus: 

(1) A = B | C = D 

(2) C = D | A = B 

Make an ABCD diagram: 



§ 428.] 



EQUIVALENTS. 



231 



AB 


Ab 


aB 


ab 




12 








CD 




12 


2 


2 


Cd 




1 






cD 




1 






cd 



Fig. 146. 

Mr. Weltcm's (1) 

Now, if A = B, except where C = D, then the combinations 
containing ABCD, AbCd, Abe, are inconsistent, and we elimi- 
nate them by making a figure 1 in those sections. 

If C = D, except where A = B, then the combinations con- 
taining ABCD, AbCd, aBCd, abCd, are inconsistent, and we 
eliminate them by making a figure 2 in those sections. 

Mr. Welton's (2) can be stated thus: 
(1) Ab = AbCD 

Make an ABCD diagram : 



AB 


Ab 


aB 


ab 












CD 




1 






Cd 




1 






cD 




1 






cd 



Fig. 147. 

Mr. Welton's (2). 

Now, if Ab = AbCD, then the combinations containing 
AbCd, Abe, are inconsistent, and we eliminate them by making 

a figure 1 in those sections. 

The Reasoning Frames now show that (1) and (2) arc not 
equivalents. But (2) is an inference from (1). 



232 



HYPOTHETICAL PROPOSITIONS. 



[ Chap. 15. 



Mr. Welton's (3) can be stated thus: 

(1) No A = B | C = D 

(2) No C = D | A = B 
Make an ABCD diagram: 



AB 


Ab 


aB 


ab 






2 


2 


2 


CD 


1 








Cd 


1 








cD 


1 








cd 



Fig. 148. 

Mr. Welton's (3) 

Now, if No A = B, except where C = D, then the combina- 
tions containing ABCd, ABc, are inconsistent, and we elimi- 
nate them by making a figure 1 in those sections. 

Again, if No C === D, except where A = B, then the 
combinations containing AbCD, aBCD, abCD, are inconsistent, 
and we eliminate them by making a figure 2 in those sections. 

Now, by combining (1) and (3) we can learn whether they are 
contradictories, for, if they are contradictories, they will cause 
the total elimination of some letter-term. 

We will put the figure 1 in those sections which are elimi- 
nated by (1) and the figure 3 in those sections which are elimi- 
nated by (3). 

Make an ABCD diagram: 



AB 


Ab 


aB 


ab 




1 


3 


3 


3 


CD 


3 


1 


1 


1 


Cd 


3 


1 






cD 


3 


1 






cd 



Fig. 149. 



§428.] 



EQUIVALENTS. 



233 



Mr. Welton's (1) and (3) combined. 

The Reasoning Frame now shows that all the A's and C'h 
are eliminated, and this proves that (1) and (3) are contradic- 
tories. 

Mr. Welton's (4) can be stated thus: 
(1) Ab = AbCd 

Make an ABCD diagram: 



AB 


Ab 


aB 


ab 






1 






CD 










Cd 





1 
1 






cD 






cd 



Fig. 150. 

Mr. Welton's (4). 

Now, if Ab = AbCd, then the combinations containing 
AbCD, Abe are inconsistent and we eliminate them by making 
a figure 1 in those sections. 

Now, by combining (2) and (4) we can learn whether they are 
contradictories. We will put a fig'ure 2 in those sections which 
Mr. Welton's (2) eliminated and a figure 4 in those sections 
which his (4) eliminated. 

Make an ABCD diagram: 



AB 


Ab 


aB 


ab 






4 






CD 




2 






Cd 




2 

4 






cD 




2 

4 




cd 



Fig. 151. 



234 



HYPOTHETICAL PROPOSITIONS. 



[ Chap. 15. 



Mr. Welton's (2) and (4) combined. 

The Seasoning Frame now shows that (2) and (4) are not con- 
tradictories, because no letter-term has been eliminated. But 
they are inconsistent because neither one can be read in the 
Reasoning Frame. This will be explained later. 
They say A = B, No A = b, 

Mr. Welton's (5) can be stated thus: 

(1) A = B, | C = d 

(2) O = d, | A = B. 

Make an ABCD diagram: 



AB 


Ab 


aB 


ab 






21 


2 


2 


CD 


21 








Cd 




1 







cD 




1 




cd 



Fig. 152. 

Mr. Welton's (5). 

Now, if A = B, except where C = d, then the combinations 
containing ABCd, AbCD, Abe, are inconsistent, and we elimi- 
nate them by making a figure 1 in those sections. 

If C = d, except where A = B, then the combinations con- 
taining ABCd, AbCD, aCD, are inconsistent, and we eliminate 
them by making a figure 2 in those sections. 

The Reasoning Frames now show that (3) and (5) are not 
equivalents. 

Let us now take Mr. Welton's (a). 

This pen is either cross-nibbed or corroded by the ink. 

Let A = this pen, 

B — cross-nibbed, 

C = corroded by the ink. 

The premise can be stated thus : 

(1) A == ABc | AbC 



428.] CONTRADICTORIES. 

Make an ABC diagram: 



235 



AB 


Ab 


aB 


ab 




1 








C 




1 






c 



Fig. 153. 
Mr. Welton's (a). 

Now, if A = ABc | AbC, then the combinations ABC, Abe, 
are inconsistent, and we eliminate them by making a figure 1 
in those sections. 
Mr. Welton's (b). 

This pen is neither cross-nibbed nor corroded by the ink, can 
be stated thus: 

(1) A = Abo 

Make an ABC diagram: 



AB 


Ab 


aB 


ab 




1 


1 






C 


1 








c 



Fig. 154. 

Mr. Welton's (b). 

Now, if A = Abe, then the combinations containing AB, 
AbC, are inconsistent, and we eliminate them by making a fig- 
ure 1 in those sections. 

Now, if (a) and (b) are contradictories, we can ascertain that 
fact by combining the two propositions. We will put a in 
the sections which (a) eliminates, and b in the sections which 
(b) eliminates. 



236 HYPOTHETICAL PROPOSITIONS. 

Make an ABC diagram: 



[ Chap. 15. 



AB 


Ab 


aB 


ab 




b 
a 


b 






C 


b 


a 






c 



Fig. 155. 

Mr. Welton's (a) and (b) combined. 

The result proves that (a) and (b) are contradictories because 
all the A's are eliminated. 

Mr. Welton's (c). 

If this pen is not cross-nibbed, it is corroded by the ink, can 
be stated thus: 

(1) Ab = AbO 

Make an ABC diagram: 



AB 


Ab 


aB 


ab 

















1 






c 



Fig. 156. 

Mr. Welton's (c). 

Now, if Ab = AbC, then the combination Abe is inconsis- 
tent, and we eliminate it by making a figure 1 in that section. 

The Reasoning Frame now shows that (a) and (c) are not 
equivalent. But (c) is an inference from (a). 

Mr. Welton's (d). 

If this pen is not cross-nibbed, it is not corroded by the ink, 

can be stated thus: 

(1) Ab = Abe 



§ 428.] CONTRADICTORIES 

Make an ABC diagram: 



237 



AB 


Ab 


aB 


ab 






1 






C 










c 



Fig. 157. 

Mr. Welton's (d). 

Now, if Ab = Abe, then the combination AbC is inconsistent, 
and we eliminate it by making a figure 1 in that section. 

Now, by combining (c) and (d) we can ascertain whether 
they are contradictories. 

Make c in the combination which (c) eliminates and d in 
the combination which (d) eliminates. 

Make an ABC diagram: 



AB 


Ab 


aB 


ab 






d 






C 




c 






c 



Fig. 158. 

Mr. Welton's (c) and (d). 

The result proves that they are not contradictories because 
no letter-term is eliminated. But they are inconsistent 
because neither can be read in the Reasoning Frame. This will 
be explained later. 

They say A = B 

This pen is cross-nibbed. 



238 



HYPOTHETICAL PROPOSITIONS. 



[Chap. 15. 



429. The Seasoning Frames have led to this discovery. 
Given two hypothetical with the same antecedent and incon- 
sistent consequents, their combination yields a categorical con- 
tradictory to the antecedent, thus: 

(1) If A = B, C = D 

(2) If A = B, = d yields, 

No A = B, thus: 

Make an ABCD diagram: 



AB 


Ab 


aB 


ab 




2 








CD 


1 








CM 


21 








cD 


21 








cd 



Fig. 159. 

Now, if A = B C = D, then the combinations containing 
ABCd, ABc, are inconsistent, and we eliminate them by mak- 
ing a figure 1 in those sections. 

Again, if A = B C = d, then the combinations containing 
ABCD, ABc, are inconsistent, and we eliminate them by mak- 
ing a figure 2 in those sections. 

The Reasoning Frame now shows that, 
No A = B. 



CHAPTER xv r. 

HYPOTHETICAL PROPOSITIONS CONTINUED. 

430. I have also made this discovery: 

Given two hypothetical having the same consequents and 
inconsistent antecedents, their combination yields a categori- 
cal having the subject of the consequents and the opposite 
of the predicate of the consequents for its subject, and the 
opposite of the subject of the antecedents for its predicate, 
and they also yield a categorical having the opposite of the 
subject of the consequents for its subject and the opposite of 
the subject of the antecedents for its predicate, thus: 

(1) If A = B, C = D 

(2) If A = b, C = D, yields, 

Cd = a 
c = a 

Make an ABCD diagram: 



AB 


Ab 


aB 


ab 












CD 


1 


2 






Cd 


1 


2 






cD 


1 


2 






cd 



Fig. 160. 

Now, if A = B, C = D, then the combinations containing 
ABCd, ABc, are inconsistent, and we eliminate them by mak- 
ing a figure 1 in those sections. 

Again, if A = b, C = D, then the combinations containing 
AbCd, Abe, are inconsistent, and we eliminate them by mak- 
ing a figure 2 in those sections. 



240 



HYPOTHETICAL PROPOSITIONS CONTINUED. [ Chap. 16. 



The Eeasoning Frame now shows that, 

Cd = a 
c = a 

431. Assume that disjunctive propositions of the form, 
A = B |C = D 
are reciprocating, that is, they are to be worked both ways, 
then the use of the Reasoning Frame shows that this kind of a 
disjunctive proposition is more definite than a hypothetical. 
Let (1) A = B | C = D 

(2) A = B | C = d 

(3) A = b | C == D 
(1) and (2) are contradictories. 

(1) and (3) are contradictories. 

(2) and (3) are not contradictories. 

Make an ABCD diagram: 



AB 


Ab 


aB 


ab 




1 


2 


2 


2 


CD 


2 


1 


1 


1 


Cd 




21 






cD 




21 






cd 



Fig. 161. 



Now, if A = B | C = D, then the combinations containing 
ABCD, AbCd, Abe, aCd are inconsistent and we eliminate 
them by making a figure 1 in those sections. 

Again, if A = B | C = d, then the combinations contain- 
ing ABCd, AbCD, Abe, aCD, are inconsistent and we elimi- 
nate them by making a figure 2 in those sections. 

The Reasoning Frame now shows the visible expression of 
the combination of (1) and (2) and it proves that they are con- 
tradictories because the letter C is totally eliminated. 



§ 481. ] CONTRADICTORIES. 

Make an ABCD diagram: 



241 



AB 


Ab 


aB 


ab 




1 


3 






CD 


3 


1 


3 

1 


3 
1 


Cd 


3 


1 






cD 


3 


1 






cd 



Fig. 162. 

Now, if A = B | C = D, then the combinations contain- 
ing ABCD, AbCd, Abe, aCd, are inconsistent and we elimi- 
nate them by making a figure 1 in those sections. 

Again, if A = b | C = D, then the combinations contain- 
ing AbCD, ABCd, ABc, aCd, are inconsistent and we elimi- 
nate them by making a figure 3 in those sections. 

The Beasoning Frame now shows the visible expression of 
the result of the combination of (1) and (3) and it proves that 
(1) and (3) are contradictories because the letter A is elimi- 
nated. 

Make an ABCD diagram: 



AB 


Ab 


aB 


ab 






3 
2 


2 


2 


CD 


3 
2 




3 


3 


Cd 


8 


2 






cD 


3 


2 






cd 



Fig. 163. 

Now, if A = B | C = d, then the combinations containing 
ABCd, AbCD, Abe, aCD, are inconsistent and we eliminate 
them by making a figure 2 in those sections. 
16 



242 



HYPOTHETICAL PROPOSITIONS CONTINUED. [ Chap. 16. 



Again, if A = b | C = D, then the combinations contain- 
ing AbCD, ABCd, ABc, aCd, are inconsistent and we elimi- 
nate them by making a figure 3 in those sections. 

The result shows the visible expression of the combination 
of (2) and (3) and it proves that they are not contradictories 
because no letter term has been eliminated. 

The combination yields these results : 

(1) No A = c 

(2) No C = a 

(3) a = c 

(4) = ABD | Abd 

(5) D = ABC | a 

432. If we consider disjunctive propositions of the form, 

(1) A | B = C | D, i. e., Ab | aB = Gd | cD, as Uni- 
versal, then disjunctives having the form, 

(2) A | B = G | d, i. e., Ab | aB = CD | cd 

(3) A | b = | D, i. e., AB | ab = Cd | cD, are not 
contradictories to, 

(1) A | B = | D. 

Make an ABCD diagram: 



AB 


Ab 


aB 


ab 






21 


1 
2 




CD 










Cd 




2 


2 




oD 




1 


1 




cd 



Fig. 164. 

Now, if A | B = C | D, then the combinations contain- 
ing AbCD, Abed, aBCD, aBcd, are inconsistent and we elimi- 
nate them by making a figure 1 in those sections. 

Again, if A | B = C | d, then the combinations AbCD, 
AbcD, aBCD, aBcD, are inconsistent and we eliminate them by 
making a figure 2 in those sections. 



432.] 



CONSISTENTS. 



243 



The result proves that (1) and (2) are not contradictories. 
They yield this result, 

A | B = Cd. 

Make an ABCD diagram: 



AB 


Ab 


aB 


ab 




3 


1 


1 


3 


CD 










Cd 










cD 


3 

- 


1 


1 


3 


cd 



Fig. 165. 

Now, if A | B = G | D, then the combinations AbCD, 
Abed, aBCD, aBcd, are inconsistent and we eliminate them by 
making a figure 1 in those sections. 

Again, if A | b = C | D, then the combinations containing 
ABCD, ABcd, abCD, abed are inconsistent and we eliminate 
them by making a figure 3 in those sections. 

The result proves that (1) and (3) are not contradictories, 
they yield these definitions: 

(1) C = d 

(2) D = o 

Make an ABCD diagram: 



AB 


Ab 


aB 


ab 




3 


2 


2 


3 


CD 










Cd 




2 


2 




cD 


3 






3 


cd 



Fig. 1GG. 



244 HYPOTHETICAL PROPOSITIONS CONTINUED. L Chap. 16. 

Now, if A | B = C | d, then the combinations AbCD, 
AbcD, aBCD, aBcD, are inconsistent and we eliminate them by 
making a figure 2 in those sections. 

Again, if A | b = C | D, then the combinations ABCD, 
ABcd, abCD, abed, are inconsistent and we eliminate them by 
making a figure 3 in those sections. 

The result proves that (2) and (3) are not contradictories. 

They yield these results, 

(1) A | b = | D 

(2) A | B= d 

But, if we were to regard disjunctive propositions having 
the form, 

A | B = C | D 
i. e., Ab | aB = Cd | cD 
as reciprocating propositions to be worked both ways, then 
propositions having the form, 
A | B = | d 
i. e., Ab | aB = CD | cd 

A | b = C | D 
i. e., AB | ab = Cd | cD 
would be contradictories of, 
A | B = C | D 
i. e., Ab | aB == Cd | cD 

433. Dr. Keynes says in "Formal Logic," p. 225, "It will be 
observed that on neither interpretation are If A then 0' and 
'If A then c' true contradictories. 

The propositions can be stated thus: 

(1) AB == ABCD. 

(2) AB = ABCd 

Make an ABOD diagram: * 



§ 434.] 



CONTRADICTORIES. 



245 



AB 


Ab 


aB 


ab 




2 








CD 


1 








Cd 


2 

1 








cD 


2 

1 








cd 



Fig. 167. 

Now, if AB = CD, then the combinations containing ABCd, 
ABc, are inconsistent and we eliminate them by making a fig- 
ure 1 in those sections. 

Again, if AB = Cd, then the combinations containing 
ABCD, ABc are inconsistent and we eliminate them by mak- 
ing a figure 2 in those sections. 

The Seasoning Frame now shows that the two propositions 
are not true contradictories because they do not eliminate 
any letter term. They are, however, inconsistent because 
neither one of them can now be read in the Reasoning Frame. 

434. Dr. Keynes also says, "As a concrete example we may 
take the propositions, 

(1) If this pen is not cross-nibbed it is corroded by the ink. 

(2) If this pen is not cross nibbed it is not corroded by tne 

ink." 

Let A = pen, 

B = cross-nibbed, 

=s corroded by the ink. 

The propositions can be stated thus: 

(1) Ab = AbC 

(2) Ab = Abo 

Make an ABC diagram: 



246 HYPOTHETICAL PROPOSITIONS CONTINUED. [ Chap. 16. 



AB 


Ab 


aB 


ab 






2 






C 




1 






c 



Fig. 168. 



Now, if Ab = AbC, then the combination Abe is inconsist- 
ent and we eliminate it by making a figure 1 in that section. 

Again, if Ab = Abe, then the combination AbC is incon- 
sistent and we eliminate it by making a figure 2 in that sec- 
tion. 

The Reasoning Frame now shows that the two propositions 
are not contradictories because no letter-term is eliminated. 
They are, however, inconsistent because neither one of them 
can now be read in the Reasoning Frame.. 

I quote from Dr. Keynes "Formal Logic" so often, because 
I consider it the best work there is on the old logic. 

435. In the old logic a hypothetical having the form of, 
If A is B then C is D, is called an affirmative proposition. 
If A is B then C is not-D is called a negative hypothet- 
ical. 

As I believe that all propositions are affirmative, I would 
suggest that propositions which in the old logic are called uni- 
versal affirmatives, be called identifying propositions and prop- 
ositions which in the old logic are called universal negatives, 
be called excluding propositions. But until this suggestion is 
adopted by logicians I shall use the term "negative proposi- 
tion," although protesting against it. I think "differential 
proposition" would be a good name for a particular negative 
proposition. 

When propositions are doubly universal, that is, of the form 
of Hamilton's U: 



§§ 436, 437.] 



AN EXAMPLE. 



247 



A is B and B is A, 
I suggest that such propositions be called reciprocating prop- 
ositions. 

436. Unless there is something to indicate to the contrary, 
I consider identifying propositions to be universal. 

Take the hypothetical, 

If A is B, C is D. and I think it means 
If All A is some B, then All C is some D. 

437. Let us take this example, 

(1) If patience is a virtue, there are painful virtues. 
I assume that this means, 
If patience is a virtue, then painful virtues exist 
Let A = patience, 
B = virtue, 
G = painful, 
D = exist. 
The proposition can be stated thus: 
AB = ABCD. 

Make an ABCD diagram: 



AB 


Ab 


aB 


ab 












CD 


1 
1 








Cd 








cD 


1 








cd 



Fig. 169. 



!Now, if AB = ABCD, then the combinations containing 
ABCd, ABc, are inconsistent and we eliminate them by making 
a figure 1 in those sections. 

The Reasoning Frame now shows the visible expression of 
the proposition, 

If patience is a virtue, then painful virtues exist. 



243 



HYPOTHETICAL PROPOSITIONS CONTINUED. [ Chap. 10. 



438. Let us take this example, 

(1) If a righteous God exists, then the wicked will not 

escape their just punishment. 

(2) If the wicked escape their just punishment, a 

righteous God does not exist. 

Let A = a righteous God, 
B = exists, 
= wicked, 
D = escape their just punishment. 

The propositions can be stated thus : 

(1) AB = ABCd 

(2) CD = ODAb 

Make an ABOD diagram: 



AB 


Ab 


aB 


ab 




1 

2 




2 


2 


CD 











Cd 


1 






cD 


1 








cd 



Fig. 170. 

Now, if AB = Cd, then the combinations containing ABCD, 
ABc, are inconsistent and we eliminate them by making a 
figure 1 in those sections. 

Again, if CD = Ab, then the combinations containing 
ABCD, aCD, are inconsistent and we eliminate them by making 
a figure 2 in those sections. 

The Reasoning Frame shows that the two propositions are 
consistent, because both of them can be read in the Reasoning 
Frame. 

439. Let us take this example, 

(1) If A is true, then C is not true, 

(2) If C is true, then A is not true, 



§440.] 



CONSISTENTS. 



249 



The premises can be stated thus: 

(1) AB = ABCd 

(2) CD = CDAb 

Make an ABCD diagram: 



AB 


Ab 


aB 


ab 




21 




2 


2 


CD 










Cd 


1 








cD 


1 








cd 



Fig. 171. 

Now, if AB = Cd, then the combinations containing ABCD, 
ABc, are inconsistent and we eliminate them by making a fig- 
ure 1 in those sections. 

Again, if CD = Ab, then the combinations containing 
ABCD, aCD, are inconsistent and we eliminate them by mak- 
ing a figure 2 in those sections. 

The Reasoning Frame now shows that these propositions are 
consistent. 

440. Let us take the following example, 

(1) If A = B then C = D 

(2) If E = F then A = B, therefore, 
If E = F C = D 

The propositions can be stated thus: 

(1) AB = ABCD 

(2) EF = EFAB 



250 HYPOTHETICAL PROPOSITIONS CONTINUED. [Chap. 16. 

Make an ABCDEF diagram: 



ABC 


ABc 


AbC 


Abe 


aBC 


aBc 


abC 


abc 








2 


2 


2 


2 


2 


2 


DEF 


















DEf 


















DeF 


















Def 


1 




2 


2 


2 


2 


2 


2 


dEF 


1 
















dEf 


1 
















deF 


1 
















def 



Fig. 172. 

Now, if AB = CD, then the combinations containing ABCd, 
ABc, are inconsistent and we eliminate them by making a 
figure 1 in those sections. 

Again, if EF = AB, then the combinations containing 
EFAb, EFa, are inconsistent and we eliminate them by making 
a figure 2 in those sections. 

The Seasoning Frame now shows that, 
If E = F then C = D. 

441. Let us take the following example: 

(1) If water is salt it will not boil at 212 degrees, 

(2) Sea water is salt, therefore, 

Sea water will not boil at 212 degrees. 
I assume that sea water is water, 
Let A = water, 
B = salt, 

C = boil at 212 degrees, 
D = sea water. 
The propositions can be stated thus: 



§441 



AN EXAMPLE. 

(1) AB = ABc 

(2) D = DB 

(3) D = DA 



251 



Make an ABCD diagram: 



AB 


Ab 


aB 


ab 




1 


2 


3 


3 
2 


CD 


I 








Cd 




2 


3 


3 
2 


cD 










cd 



Fig. 173. 

Now, if AB = ABc, then the combinations containing ABC 
are inconsistent and we eliminate them by making a figure 1 
in those sections. 

Again, if D = DB, then the combinations containing Db are 
inconsistent and we eliminate them by making a figure 2 in 
those sections. 

Again, if D = DA, then the combinations containing Da 
are inconsistent and we eliminate them by making a figure 3 
in those sections. 

From the uneliminated combinations we can get the follow- 
ing definitions: 

(1) D = Dc, which can be translated, 
Sea water will not boil at 212 degrees. 

(2) AB = ABc, which can be translated, 
Salt water will not boil at 212 degrees. 

From the eliminated combinations we can get the following 
definitions: 

(1) No D = C, which can be translated, 
No sea water will boil at 212 degrees. 

(2) No AB = C, which can be translated, 
No salt water will boil at 212 degrees. 



252 



HYPOTHETICAL PROPOSITIONS CONTINUED. [ Chap. 16. 



442. Let us take the following example : 

(1) Whenever is D then E is F, 

(2) Whenever A is B then C is D, therefore, 
Whenever A is B, E is F. 

The propositions can be stated thus: 

(1) CD = CDEF 

(2) AB = ABCD 

Make an ABCDEF diagram: 



ABC 


ABc 


AbC 


Abe 


aBC 


aBc 


abC 


abc 






2 














DEF 


1 


2 


1 




1 




1 




DEf 


1 


2 


1 




1 




1 




DeF 


1 


2 


1 




1 




1 




Def 


2 


2 














dEF 


2 


2 
2 














dEf 


2 














deF 


2 


2 














def 



Fig. 174. 



Now, if CD = EF, then the combinations containing CDEf, 
CDeF, CDef, are inconsistent and we eliminate them by making 
a figure 1 in those sections. 

Again, if AB = CD, then the combinations containing 
ABCd, ABc, are inconsistent and we eliminate them by making 
a figure 2 in those sections. 

The Reasoning Frame now shows that, 
Whenever A = B, E = F. 

This example is a syllogism in Barbara, taken from Dr. 
Keynes' "Formal Logic," p. 300. 



§443.] 



AN EXAMPLE IN DARAPTI. 



253 



443. Let us take the following example in Darapti: 

(1) "Whenever C is D, E is F, 

(2) "Whenever C is D, A is B, therefore, 

Sometimes when A is B, E is F." 

The propositions can be stated thus: 

(1) CD = CDEF 

(2) CD = CDAB 

Make an ABCDEF diagram: 



ABC 


ABc 


AbC 


Abe 


aBC 


aBc 


abC 


abc 








2 




2 




2 




DEF 


1 




21 




21 




21 




DEf 


1 




21 




21 




21 




DeF 


1 




21 




21 




21 




Def 


















dEF 


















dEf 


















deF 


















def 



Fig. 175. 

Now, if CD = EF, then the combinations containing CDEf, 
CDeF, CDef, are inconsistent and we eliminate them by mak- 
ing a figure 1 in those sections. 

Again, if CD = AB, then the combinations containing 
CDAb, CDaB, CDab, are inconsistent and we eliminate them 
by making a figure 2 in those sections. 

The Keasoning Frame now shows that we can get this defi- 
nition of AB, 

AB = ABEF | ABEf | ABeF | ABef, which the old logic 
would translate, 



254 



HYPOTHETICAL PROPOSITIONS CONTINUED. [ Chap. 16. 



Sometimes when A is B, E is F, 
but I consider this too positive a statement to make. The 
Reasoning Frame leaves the matter in doubt. 

444. Let us take this example : 

(1) Never when C is D is it the case that A is B. 

(2) Whenever E is F, C is D, therefore, 

Never when E is F, is it the case that A is B, there- 
fore, 

Never when A is B, is it the case that E is F. 
The propositions can be stated thus: 

(1) No CD = AB 

(2) EF — EFCD. 

Make an ABCDEF diagram: 



A.B( 


ABc 


AbC 


Abe 


aBC 


a Be 


abC 


a be 




1 


2 




2 




2 




2 


DEF 


1 
















DEf 


1 
















DeF 


1 
















Def 


2 


2 


2 


2 


2 


2 


2 


2 


dEF 


















dEf 


















deF 


















def 



Fig. 176. 



Now, if No CD = AB, then the combinations containing 
ABCD are inconsistent and we eliminate them by making a 
figure 1 in those sections. 

Again, if EF === CD, then the combinations containing EFc, 
EFd, are inconsistent and we eliminate them by making a 
figure 2 in those sections. 



§§ 445, 446.] 



MODUS TOLLENS. 



255 



The Seasoning Frame now shows that we can draw the fol- 
lowing conclusions: 

(1) Never when E is F is it the case that A is B, 

(2) Never when A is B is it the case that E is F. 

(Keynes' "Formal Logic," p. 303.) 

445. Let us take the following example: 

(1) If A = B, C = D, 

(2) C = d, therefore, A = b. 
The premises can be stated thus: 

(1) AB = ABCD 

(2) C == Cd. 

Make an ABCD diagram: 



AB 


Ab 


aB 


ab 




2 


2 


2 


2 


CD 


1 








Cd 


1 








cD 


1 








cd 



rig. 177. 

Now, if AB = CD, then the combinations containing ABCd, 
ABc are inconsistent and we eliminate them by making a 
figure 1 in those sections. 

Again, if C = d, then the combinations containing CD are 
inconsistent and we eliminate them by making a figure 2 in 
those sections. 

The Seasoning Frame now shows that, 

A = Ab. 
The above is an example in the modus fallens, (also called 
the destructive hypothetical syllogism.) 

(Dr. Keynes' "Formal Logic," p. 304.) 
446. Let us take the following example: 
(1) If A = b then C = D, 



256 



HYPOTHETICAL PROPOSITIONS CONTINUED. [ Chap. 16. 



(2) 



= d, therefore, 
A = B. 
The premises can be stated thus: 

(1) Ab = AbCD 

(2) C = Cd. 

Make an ABCD diagram: 



AB 


Ab 


aB 


ab 




2 


2 


2 


2 


CD 




1 






Cd 




1 






cD 




1 






cd 



Fig. 178. 

Now, if Ab = CD, then the combinations containing AbCd, 
Abe, are inconsistent and we eliminate them by making a 
figure 1 in those sections. 

Again, if C = d, then the combinations containing CD are 
inconsistent and we eliminate them by making a figure 2 in 
those sections. 

The Keasoning Frame now shows that, 

A = AB. 

This example is in the modus tollens and corresponds to 
Camestres. 



447. Let us take the following example; 

(1) If A = BthenC = d 

(2) C = D, therefore, 
A = b 

The premises can be stated thus: 

(1) AB = ABCd 

(2) C = CD 

Make an ABCD diagram: 



§448.] 



AN EXAMPLE IN MODUS TOLLENS. 



257 



AB 


Ab 


aB 


ab 




1 








CD 


2 


2 


2 


2 


Cd 


1 








cD 


1 








cd 



Fig. 179. 



Now, if AB = Cd, then the combinations containing ABCD, 
ABc, are inconsistent and we eliminate them by making a 
figure 1 in those sections. 

Again, if C = D, then the combinations containing Cd are 
inconsistent and we eliminate them by making a figure 2 in 
those sections. 

The Seasoning Frame now shows that, 

A = Ab. 

This example is in the modus tollens and corresponds to 
Cesare. 

448. Let us take the following example: 

(1) If A = b, then C = d 

(2) C = D, therefore, A = B 
The premises can be stated thus : 

(1) Ab = AbCd 

(2) C = CD 

Make an ABCD diagram: 



AB 


Ab 


aB 


ab 






1 






CD 


2 


2 


2 


2 


Cd 




1 






cD 




1 






cd 



17 



Fig. ISO. 



258 HYPOTHETICAL PROPOSITIONS CONTINUED. [ Chap. 16. 

Now, if Ab = Cd, then the combinations containing AbCD, 
Abe, are inconsistent and we eliminate them by making a figure 
1 in those sections. 

Again, if = D, then the combinations containing Cd are 
inconsistent and we eliminate them by making a figure 2 in 
those sections. 

The Seasoning Frame now shows that A = AB. 

449. Let us take the following example: 

(1) If A = B then C = D 

(2) C = D, therefore, 
A = B. 

The premises can be stated thus: 

(1) AB = ABOD 

(2) = CD 

Make an ABCD diagram: 



AB 


Ab 


aB 


ab 












CD 


21 


2 


2 


2 


Cd 


1 








cD 


1 








cd 



Fig. 181. 

Now, if AB = CD, then the combinations containing ABCd, 
ABc, are inconsistent and we eliminate them by making a 
figure 1 in those sections. 

Again, if C = D, then the combinations containing Cd are 
inconsistent and we eliminate them by making a figure 2 in 
those sections. 

The Reasoning Frame now shows that, 
A = B | b 

Hence, it is a fallacy to regard the affirmation of the conse- 
quent as justifying the affirmation of the antecedent. 



§ 450.] 



AN EXAMPLE. 



25.9 



450. Let us take the following pair of hypothetical syllo- 
gisms : 

(1) If A = B then C = D 

(2) C = d, therefore, A = b 

(3) If C = d then A = b 

(4) = d, therefore, A = b. 

The premises can be stated thus : 

(1) AB = ABCD 

(2) C = Cd 

(3) Cd = CdAb 

(4) C == Cd 

Make an ABCD diagram: 



AB 


Ab 


aB 


ab 




4 
2 


4 

2 


4 

2 


4 
2 


CD 


31 




3 


3 


Cd 


1 








cD 


1 








cd 



Fig. 182. 



Now, if AB = CD, then the combinations containing ABCd, 
ABc, are inconsistent and we eliminate them by making a 
figure 1 in those sections. 

Again, if C = d, then the combinations containing CD are 
inconsistent and we eliminate them by making a figure 2 in 
those sections. 

Again, if Cd = Ab, then the combinations containing ABCd, 
aCd, are inconsistent and we eliminate them by making a 
figure 3 in those sections. 

Again, if C = d, then the combinations containing CD are 
inconsistent and we eliminate them by making a figure 4 in 
those sections. 

The Reasoning Frame now shows that the given syllogisms 



260 HYPOTHETICAL PROPOSITIONS CONTINUED. [Chap. 16. 

are not contradictories because no letter-term is eliminated 
and it also shows that their combination yields the conclusions, 

A == Ab 

C = Cd 

EXEROISES. 

451. (1) What conclusions can be drawn from the follow- 
ing pair of hypothetical propositions, 
If A = b then A = c 
B = c? 

(2) What conclusions can be drawn from the following pair 
of propositions, 

If a = b then a = c 
a = C? 

(3) What conclusions can be drawn from the following pair 
of propositions, 

If aB = aBcthena = c 
a == aB? 

(4) What conclusions can be drawn from the following pair 
of propositions, 

If a = b then = D 
a = b? 

(5) What conclusions can be drawn from the following pair 
of propositions, 

If A = B then c = d 
c = D? 

(6) What conclusions can be drawn from the following pair 
of propositions, 

If A = AC then A == AB 
If A = AC then A = b? 

(7) What conclusions can be drawn from the following pair 
of propositions, 

If A = Ab then D = De 
If A = AbthenC = CE? 

(8) What conclusions can be drawn from the following pair 
of propositions, 

If A = AC then B = Bd 



§451.] EXERCISES. 261 

If A = AC then No A = d? 

(9) What conclusions can be drawn from the following pair 
of propositions, 

If A = b then C = d 
If E = f then A = b? 

(10) What conclusions can be drawn from the following pair 
of propositions, 

If No A = B then C = D 
C = d? 

(11) What conclusions can be drawn from the following pair 
of propositions, 

Never when A = B is it the case that G = D 
No A = B? 



CHAPTER XVII. 

DILEMMAS. 

452. Dr. Keynes, a most accomplished logician, in "Formal 
Logic," p. 316, says : "The proper place of the dilemma 
amongst hypothetical and disjunctive arguments is difficult to 
determine, inasmuch as conflicting definitions are given by 
different logicians. 

The following definition may be taken, perhaps, as on the 
whole the most satisfactory: 

A dilemma is a formal argument containing a premise in 
which two or more hypotheticals are conjunctively affirmed, 
and a second premise in which the antecedent of these hypo- 
theticals are alternately affirmed or their consequents alter- 
nately denied ." 

453. Prof. Bain in his "Deductive and Inductive Logic," p. 
121, defines a dilemma thus: 

"The dilemma contains a conditional and a disjunctive pro- 
position. If the antecedent of a conditional is made disjunc- 
tive, there emerges, what Whately calls a 'Simple Constructive 
Dilemma.' He gives this example: 
"If either A or B is, C is, 
Now, either A or B is, 
Therefore, C is." 

454. Let us take this example: 

(1) If the barometer falls then there will be either wind or 
rain, 

(2) The barometer is falling, therefore, there will be either 
wind or rain. 

Let A = barometer, 
B = falls, 
C = wind, 
D — rain. 



§ 455.] 



AN EXAMPLE. 



263 



The premises can be stated thus: 

(1) AB = Cd | cD 

(2) A == AB 

Make an ABCD diagram: 



AB 


Ab 


aB 


ab 




1 


2 






CD 




2 






Cd 




2 






cD 


1 


2 






cd 



Fig. 183. 



Now, if AB = C | D, then the combinations ABCD, ABcd, 
are inconsistent, and we eliminate them by making a figure 1 in 
those sections. 

Again, if A = B, then the combinations containing Ab are 
inconsistent, and we eliminate them by making a figure 2 in 
those sections. 

From the uneliminated combinations we can get this defi- 
nition: 

A = BCd | BcD, which can be translated, 

The barometer is falling and there will be wind or rain. 

455. Let us take this example: 

(1) If the barometer falls then there will be either wind or 
rain. 

(2) It is not raining. 

The premises can be stated thus: 

(1) AB = C | D 

(2) AB = ABd 

Make an ABCD diagram: 



264 



DILEMMAS. 



[ Chap. 17. 



AB 


Ab 


aB 


ab 




21 








CD 










Cd 


2 








cD 


1 








cd 



Fig. 184. 

Now, if AB 3=x | D, then the combinations containing 
ABOD, ABcd, are inconsistent, and we eliminate them by mak- 
ing a figure 1 in those sections. 

Again, if AB = d, then the combinations containing ABD are 
inconsistent, and we eliminate them by making a figure 2 in 
those sections. 

From the uneliminated combinations in the Reasoning 
Frame we can get this definition, 

ABd = ABdC, which can be translated, 

If the barometer is falling and there is no rain, then there is 
wind. 

456. Let us take this example: 

(1) If the barometer falls then there will be either wind or 
rain. 

(2) It is raining. 

The premises can be stated thus: 

(1) AB = | D 

(2) AB = ABD 



Make an ABOD diagram: 



§ 457.] 



AN EXAMPLE. 



AB 


Ab 


aB 


ab 




1 








CD 


2 








Cd 










cD 


2 
1 








cd 



265 



Fig. 185. 

Now, if AB a=C | D, then the combinations ABCD, ABcd, 
are inconsistent, and we eliminate them by making a figure 1 in 
those sections. 

Again, if AB = D, then the combinations containing ABd are 
inconsistent, and we eliminate them by making a figure 2 in 
those sections. 

From the uneliminated combinations we can get this defi- 
nition : 

ABD = ABcD, which can be translated, 

If the barometer is falling and there is rain then there is no 
wind. 

457. Let us take this example: 

(1) If the barometer falls then there will be either wind or 
rain, 

(2) There is wind. 

The premises can be stated thus: 

(1) AB = C | D 

(2) AB = ABO 

Make an ABCD diagram: * 



2G6 



DILEMMAS. 



[Chap. 17* 



AB 


Ab 


aB 


ab 




1 








CD 










Cd 


2 








cD 


2 

1 








cd 



Fig. 186. 

Now, if AB = G | D, then the combinations ABCD, ABcd, 
are inconsistent, and we eliminate them by making a figure 1 in 
those sections. 

Again, if AB = C, then the combinations containing ABc 
are inconsistent, and we eliminate them by making a figure 2 in 
those sections. 

From the uneliminated combinations we can get this defi- 
nition : 

ABO = ABCd, which can be translated, 

If the barometer is falling and there is wind, then there is no 
rain. 



458. Let us take this example: 

(1) If the barometer falls then there will be either wind or 
rain. 

(2) There is no wind. 

The premises can be stated thus: 

(1) AB = C | D 

(2) AB = ABc 

Make an ABCD diagram: 



§ 459.] 



AN EXAMPLE. 



367 



AB 


Ab 


aB 


ab 




21 








CD 


2 








Cd 










cD 


1 








cd 



Fig. 187. 

Now, if AB == C | D, then the combinations ABCD, ABcd, 
are inconsistent, and we eliminate them by making a figure 1 in 
those sections. 

Again, if AB = c, then the combinations containing ABC are 
are inconsistent, and we eliminate them by making a figure 2 
in those sections. 

From the uneliminated combinations we can get this defi- 
nition: 

ABc == ABcD, which can be translated, 

If the barometer is falling and there is no wind, then there is 
rain, 

459. Let us take this example: 

(1) If the barometer falls then there will be either wind or 
rain. 

(2) There is no wind, 

(3) There is no rain, 

The premises can be stated thus: 

(1) AB = C | D 

(2) AB == ABc 

(3) AB = ABd 



Make an ABCD diagram: 



263 



DILEMMAS. 



[Cbap. 17, 



AB 


Ab 


aB 


ab 




21 
3 








CD 


2 








Cd 


3 








cD 


1 








cd 



Fig. 188. 

Now, if AB == G | D, then the combinations ABCD, ABc'd, 
are inconsistent, and we eliminate them by making a figure 1 in 
those sections. 

Again, if AB = c, then the combinations containing ABC are 
inconsistent, and we eliminate them by making a figure 2 in 
those sections. 

Again, if AB = d, then the combinations containing ABD are 
inconsistent, and we eliminate them by making a figure 3 in 
those sections. 

From the uneliminated combinations we can get this defini- 
tion: 

A = Ab, which can be translated, 
The barometer is not falling. 



460. Let us take this example from Prof. Bain's Logic, p. 
122. 

(1) If the orbit of a comet is diminished, either the comet 
passes through a resisting medium or the law of gravitation 
is partially suspended. 

(2) But the second alternative is inadmissible; hence, if the 
orbit of a comet is diminished, there is a resisting medium. 

Let A = orbit of a comet, 

B = diminished, , 

C = passes through a resisting medium, 
D = law of gravitation, 
E = partially suspended. 



§461.] 



AN EXAMPLE. 



269 



The premises can be stated thus 

(1) AB = ABO | ABDE 

(2) No D = E 

Make an ABODE diagram: 



AB 


Ab 


aB 


ab 




1 
2 


2 


2 


2 


CDE 










CDe 










CdE 










Cde 


2 


2 


2 


2 


cDE 


1 








cDe 


1 








cdE 


1 








cde 



Fig. 189. 

Now, if AB = C | DE, then the combinations containing 
ABODE, ABcDe, ABcd, are inconsistent, and we eliminate 
them by making a figure 1 in those sections. 

Again, if No D = E, then the combinations containing DE 
are inconsistent, and we eliminate them by making a figure 2 in 
those sections. 

The Keasoning Frame now shows that we can get this defini- 
tion of AB: 

AB = ABC, which can be translated, 

If the orbit of a comet is diminished, there is a resisting 
medium. 



461. Let us take this example: 

(1) If a classical education is worth the cost, then either it 
must be pre-eminently fitted to develop the mental powers, or 
it must furnish exceedingly valuable information. 



270 



DILEMMAS. 



[ Chdp. 17. 



(2) But neither alternative can be maintained; hence, a class- 
ical education is not worth the cost. 

Let A = classical education, 
B = worth the cost, 

= pre-eminently fitted to develop the mental powers, 
D = furnish exceedingly valuable information, 

The premises can be stated thus: 

(1) AB = ABO | ABD 

(2) A = Ac 

(3) A = Ad 
Make an ABCD diagram: 



AB 


Ab 


aB 


ab 




3 
21 


3 






CD 


2 


2 






Cd 


3 


3 






cD 


1 








cd 



Fig. 190. 

Now, if AB = C | D, then the combinations ABCD, ABcd, 
are inconsistent, and we eliminate them by making a figure 1 
in those sections. 

Again, if A = c, then the combinations containing AC are 
inconsistent, and we eliminate them by making a figure 2 in 
those sections. 

Again, if A = d, then the combinations containing AD are 
inconsistent, and we eliminate them by making a figure 3 in 
those sections. 

From the uneliminated combinations we can get this defini- 
tion of A: 

A = Ab, which can be translated, 

A classical education is not worth the cost. 

4G2. Let us take this example: 

(1) If schoolmasters can claim exemption from Poor's rates, 
then it must be either by statute or by the common law. 



8 462.] 



AN EXAMPLE. 



271 



(2) Now no statute exempts them. 

(3) And the common law does not apply; hence they can 
claim no exemption from Poor's rates. 

Let A = schoolmaster, 

B = claim exemption from Poor's rates, 

= statutes, 

D = the common law. 
The premises can be stated thus : 

(1) AB = ABC | ABD 

(2) No C = AB 

(3) No D = AB 

Make an ABOD diagram: 



AB 


Ab 


aB 


ab 




21 
3 








CD 


2 








Cd 


3 








cD 


1 








cd 



Fig. 191. 

Now, if AB = C | D, then the combinations ABCD, ABcd, 
are inconsistent, and we eliminate them by making a figure 1 in 
those sections. 

Again, if No C = AB, then the combinations containing ABO 
are inconsistent, and we eliminate them by making a figure 2 
in those sections. 

Again, if No D = AB, then the combinations containing 
ABD are inconsistent, and we eliminate them by making a fig- 
ure 3 in those sections. 

From the uneliminated combinations we can get this defini- 
tion of A : 

A = Ab, which can be translated, 

Schoolmasters can claim no exemption from Poor's rates. 
(Bain's "Deductive and Inductive Logic," p. 122.) 



272 



DILEMMAS. 



[ Chap. 17. 



463. Let us take this example from Smart's "Manual of 
Logic," p. 169: 

(1) If Aeschines joined in the public rejoicings, then he is 
inconsistent. 

(2) If he did not, then he is unpatriotic. 

(3) But he either joined or not. 

Therefore, either he is inconsistent or unpatriotic, or he is 
both inconsistent and unpatriotic. 

Let A = Aeschines, 

B = joined in the public rejoicings, 

= inconsistent, 

D = unpatriotic. 
The premises can be stated thus : 

(1) AB = ABC. 

(2) Ab = AbD 

(3) A = B | b 

Make an ABCD diagram: 



AB 


Ab 


aB 


ab 












CD 




2 






Cd 


1 








cD 


1 


2 






cd 



Fig. 192. 



Now, if AB = C, then the combinations containing ABc are 
inconsistent, and we eliminate them by making a figure 1 in 
those sections. 

Again, if Ab = D, then the combinations containing Abd 
are inconsistent, and we eliminate them by making a figure 2 

in those sections. 

The proposition A = B | b, has no logical force because It 



§ 464.] 



AN EXAMPLE. 



273 



denies nothing, and it does not cause us to eliminate any com- 
bination. 

From the uneliminated combinations we can get this defini- 
tion of A: 

A = Cd | cD | CD, which can be translated, 

Aeschines was inconsistent or unpatriotic, or both. 

464. Let us take this example: 

(1) If a science furnishes useful facts it is worthy of being 
cultivated. 

(2) If the study of it exercises the reasoning powers, it is 
worthy of being cultivated. 

(3) But either a science furnishes useful facts, or its study 
exercises the reasoning powers, therefore it is worthy of being 
cultivated. 

Let A = the study of a science, 

B = useful facts, 

C = worthy of being cultivated, 

D = exercises the reasoning powers. 
The premises can be stated thus : 

(1) AB = ABO 

(2) AD = ADO 

(3) A = B | D 

Make an ABOD diagram: 



AB 


Ab 


aB 


ab 




3 








CD 




3 






Cd 


3 

21 


2 






cD 


1 


3 






cd 



Fig. 193. 

Now, if AB = C, then the combinations containing A Be are 
inconsistent, and we eliminate them by making a figure 1 in 
those sections. 
18 



274 DILEMMAS. [Chap. 17. 

Again, if AD == C, then the combinations containing ADc 
are inconsistent, and we eliminate them by making a figure 2 in 
those sections. 

Again, if A = B | D, then the combinations containing ABD, 
Abd are inconsistent, and we eliminate them by making a figure 
3 in those sections. 

From the uneliminated combinations we can get this defini- 
tion of A : 

A = AC, which can be translated, 

The study of science is worthy of being cultivated. 

465. Let us take this example : 

(1) If this man were wise, he would not speak irreverently 
of scripture in jest. 

(2) If he were good, he would not do so in earnest. 

(3) But he does it either in jest or earnest, 

Therefore he is either not wise or not good. 

Let A = this man, 
B = wise, 

= speak irreverently of scripture in jest, 
D = good, 
E = speak irreverently of scripture in earnest. 

The premises can be stated thus: 

(1) AB = ABc 

(2) AD = ADe 

(3) A = C | E 



§466.] A SIMPLE DESTRUCTIVE DILEMMA 

Make an ABODE diagram: 



275 



AB 


Ab 


aB 


ab 




21 
3 


32 






CDE 


1 








CDe 


31 


3 






CdE 


1 








Cde 


2 


2 






cDE 


3 


3 






cDe 










cdE 


3 


3 






cde 



Fig. 194. 

Now, if AB = c, then the combinations containing ABC are 
inconsistent, and we eliminate them by making a figure 1 in 
those sections. — - 

Again, if AD = e, then the combinations containing ADE 
are inconsistent, and we eliminate them by making a figure 2 in 
those sections. 

Again, if A = C | E, then the combinations containing ACE, 
Ace, are inconsistent, and we eliminate them by making a fig- 
ure 3 in those sections. 

From the uneliminated combinations we can get this defini- 
tion of A: 

A == bCDe | bCde | BcdE | bcdE, which can be trans- 
lated, 

This man is either good and not wise, or not wise and not 
good, or wise and not good. 

466. On p. 317, Dr. Keynes gives the following example of 
a Simple Destructive Dilemma. 

If A is B, C is D, and if A is B, E is F, 

But either C is not D, or E is not F, 
Therefore, A is not B. 



276 



DILEMMAS. 



[ Chap. 17. 



The premises can be stated thus: 

(1) AB =± ABCD 

(2) AB = ABEF 

(3) = d | E = 

Make an ABCDEF diagram: 



ABC 


ABc 


AbC 


Abe 


aBO 


aBc 


abC 


abc 




3 


1 


3 




3 




3 




DEF 


2 


12 














DEf 


32 


12 


3 




3 




3 




DeF 


32 


21 


3 




3 




3 




Def 


1 


13 




3 




3 




3 


dEF 


13 
2 


1 
2 


3 




3 




3 




dEf 


1 
2 


1 
2 














deF 


1 
2 


1 
2 














def 



Fig. 195. 



Now, if AB = CD, then the combinations containing ABCd, 
ABc, are inconsistent, and we eliminate them by making a fig- 
ure 1 in those sections. 

Again, if AB = EF, then the combinations ABEf, ABe, are 
inconsistent, and we eliminate them by making a figure 2 in 
those sections. 

Again, if C = d | E = f , then the combinations containing 
CdEf, CDEF, CDe, cEF, are inconsistent, and we eliminate 
them by making a figure 3 in those sections. 

From the uneliminated combinations we can get this defiini- 
tion of A: 

A = Ab 



§ 467] 



LETTERING THE DIAGRAMS. 



277 



467. The method which I use to determine the combinations 
which the proposition = d | E = f , will cause us to elimi- 
nate it as follows: 

I make an ABCD diagram and letter it thus : I commence at 
the upper left-hand corner and over it write Cd; over the next 
file I write CD; over the next one cd, and over the last one cD. 
Against the top row I write Ef; against the second row EF; 
against the third row ef, and against the fourth row eF. 

We can always letter the files and rows with any letters we 
please, provided we follow the method of changing one letter 
at a time, beginning with the last letter. But, of course, a 
positive letter and its negative cannot occupy the same section. 

When disjunctive propositions are given to us to solve, in 
which the letters do not come in their regular order, or in which 
unusual letters, such as P, Q, R, are used, we can make the 
proper diagram and letter the files and rows as above directed, 
and then for the letters in the eliminated combinations substi- 
tute the letters A, B, C, etc., which we are accustomed to work 
with. 

This course will tend to prevent our making mistakes in 
handling complex propositions. 

Make a CdEf diagram as above directed: 



Cd 


CD 


cd 


cD 




1 
2 






Ef 




1 
2 


2 


2 


EF 




1 




ef 




I 






eF 



Fig. 196. 



Now, if = d, except where E = f , then the combinations 
containing CdEf, CDEF, CDe, are inconsistent, and we elimi- 
nate them by making a figure 1 in those sections. 



278 



DILEMMAS. 



[ Chap. 17. 



Again, if E = f , except where C = d, then the combinations 
containing CdEf, CDEF, cEF, are inconsistent, and we elimi- 
nate them by making a figure 2 in those sections. 

The Reasoning Frame now shows us the combinations which 
the proposition, 

= d ) E =*f 
will cause us to elimiuate. 

4GS. The preceding dilemma should be distinguished from 
the following hypothetical syllogism: 

(1) If A = B, C = D and E = F 

(2) But C = d and E == f 

Therefore, A = b 
The premises can be stated thus: 

(1) AB == ABCD 

(2) AB = ABEF 

(3) = Cd 

(4) E = Ef 

Make an ABCDEF diagram: 



ABC 


ABc 


AbC 


Abe 


aBC 


aBc 


abC 


abc 




42 
3 


21 

4 


4 
3 


4 


4 
3 


4 


4 
3 


4 


DEF 


3 


1 


3 




3 




3 




DEf 


23 


21 


3 




3 




3 




DeF 


3 


21 


3 




3 




3 




Def 


2 

14 


2 
41 


4 


4 


4 


4 


4 


4 


dEF 


21 


21 














dEf 


21 


21 














deF 


21 


21 














def 



Fig. 197. 



§§ 4G9, 470.] DILEMMAS USUALLY FALLACIES. 279 

Now, if AB = ABCD, then the combinations containing 
ABCd, ABc, are inconsistent, and we eliminate them by making 
a figure 1 in those sections. 

Again, if AB = EF, then the combinations containing ABEf, 
ABe, are inconsistent, and we eliminate them by making a fig- 
ure 2 in those sections. 

Again, if C = d, then the combinations containing CD are 
inconsistent, and we eliminate them by making a figure 3 ; n 
those sections. 

Again, if E = f, then the combinations containing EF are 
inconsistent, and we eliminate them by making a figure 4 in 
those sections. 

From the uneliminated combinations we can get this defini- 
tion of A : 

A = Ab. 

469. Prof. Bain, in Logic p. 122, makes this very sensible 
remark, 

"The dilemma, although occasionally a useful form, is per- 
haps oftener a snare." 

Dilemmas are usually fallacies because of a defective enum- 
eration of the different alternatives. Speakers are apt to give 
only those alternatives which they think will serve their pur- 
pose. 

470. Let us take this example: 

(1) If E = f, then A = b 

(2) If E = f, then C = d 

(3) But either A == B | C = D 

Therefore E = F 
The premises can be stated thus: 

(1) Ef = EfAb 

(2) Ef = EfCd 

(3) A =B I C = D 



280 DILEMMAS. 

Make an ABODEF diagram: 



[ Chap. 17. 



ABC 


ABc 


AbC 


Abe 


aBC 


aBc 


abC 


abc 




3 






3 










DEF 


31 
2 


12 


2 


3 
2 


12 


12 


12 


12 


DEf 


3 






3 










DeF 


3 






3 










Def 






3 


3 


3 




3 




dEF 


1 


12 


3 


3 
2 


3 
1 


12 


3 
1 


12 


dEf 






3 


3 


3 




3 




deF 






3 


3 


3 




3 




def 



Fig. 198. 

Now, if Ef = Ab, then the combinations containing Ef AB, 
Efa, are inconsistent, and we eliminate them by making a figure 
1 in those sections. 

Again, if Ef = Cd, then the combinations containing EfCD, 
Efc, are inconsistent, and we eliminate them by making a fig- 
ure 2 in those sections. 

Again, if A = B | = D, then the combinations contain- 
ing ABCD, AbCd, Abc, aCd, are inconsistent, and we elimi- 
nate them by making a figure 3 in those sections. 

From the uneliminated combinations we can get this defini- 
tion of E: 

E = EF 

471. On p. 109, Archbishop Whately in "Easy Lessons in 
Reasoning," says: 

"This kind of argument (dilemmas) was urged by the oppon- 
ents of Don Carlos the Pretender to the Spanish throne, which 
he claimed as heir-male against his neice, the Queen, by virtue 
of the Salic law, excluding females, which was established (con- 
trary to the ancient Spanish usage) by a former King of Spain, 



§ 471.] 



AN EXAMPLE. 



281 



and was repealed by King Ferdinand. They say, 'If a King of 
Spain has a right to alter the law of succession, Carlos has no 
claim; and if no King of Spain has that right, Carlos has no 
claim; but a King of Spain either has or has not such right; 
therefore (on either supposition) Carlos has no claim.' " 
Let A = King of Spain, 

B = right to alter the law of succession, 

C = Carlos, 

D = claim. 
The propositions can be stated as follows: 

(1) AB = ABCd 

(2) Ab = AbCd 

(3) A == B | b 

Make an ABCD diagram: 



AB 


Ab 


aB 


ab 




1 


2 






CD 










Cd 


1 


2 






cD 


1 


2 






cd 



Fig. 199. 



Now, if AB = Cd, then the combinations containing ABCD, 
ABc, are inconsistent, and we eliminate them by making a fig- 
ure 1 in those sections. 

Again, if Ab = AbCd, then the combinations containing 
AbCD, Abe, are inconsistent, and we eliminate them by mak- 
ing a figure 2 in those sections. 

The proposition A = B | b has no predicative force, and 
does not cause the elimination of any combination. 

From the uneliminated combinations we can get these defini- 
tions: 



8$ 



DILEMMAS. 



[ Chap. 17. 



(1) AB = ABCd 

(2) Ab = AbCd 

These propositions can be translated thus: 
. (1) If a King of Spain had a right to alter the law of succes- 
sion, then Carlos has no claim. 

(2) If a King of Spain has not that right, then Carlos has no 
claim. 

472. All the following disjunctive propositions are equiva- 
lents, because each causes us to eliminate the same combina- 
tions: 



(1) 
(2) 

(3) B | 

(4) Ab 

(5) Be 

(6) A 

(7) Ac 

(8) A 

(9) Ad 

(10) A | 

(11) AD 

(12) A 

(13) A 

(14) AC 

(15) a | 



A = B 
AB I ab 



C = 

| aB = 

| bC = 

C = 

| aC 

D = 

| aD 

d == 

| ad 

b = 

c = 

| ac 

B = 



(16) ab | AB 



C = D 
= CD | cd 

A | B 
= Cd | cD 
= Ab | aB 

B | D 
= Bd | 
= B | C 
= Be 

B | c 
= BC 

C | d 

B | d 
== BD 

c | D 
== cd ! 



bD 



bC 



bo 



bd 



CD 



The eliminated combinations will be, 

ABcD, ABCd, AbCD, Abed, aBCD, aBcd, abCd, abcD 



EXERCISES. 

473. (1) What conclusions can be drawn from the following 
propositions? 

If A | B = C, then D = E 
A | B = C 



§ 473.] EXERCISES. 283 

(2) What conclusions can be drawn from the following pair 
of propositions? 

If a = d, then B | C = E 
0= E 

(3) What conclusions can be drawn from the following propo- 
sitions? 

If A = B, then A = 
If C = D, then A = D 
C = D 

(4) What conclusions can be drawn from the following pair 
of propositions? 

If B = A, then D = C|F = E 
B == A 

(5) What conclusions can be drawn from the following propo- 
sitions? 

If B = A, then D = 
If B = A, then F = E 

D = c|F-e 

(6) What conclusions can be drawn from the following propo- 
sitions? 

If E = F, then A — B 
If E = F, then C = D 
A = b I C = d 



CHAPTER XVIII. 

STATING PROPOSITIONS. 

474. I think that the beginner in our system of logic is 
more likely to have trouble in stating propositions than in 
doing any other part of the work. This arises from the fact 
that the English language is very prolific in expressions which 
substantially assert the same idea in many different ways. 

In order to assist the reader in stating propositions, I will 
give some examples of the manner in which terms and propo- 
sitions should be stated. 

In the first line we will state the term or proposition, and 
under it we will put its symbolic expression, 
Not- A, not any A, 

a. 
A not-B, A except B, A omitting B, 
A excluding B, A neglecting B, A without B, A setting 
aside B, A barring B, 

Ab. 
A and B, A also B, A along with B, 
A likewise B, A moreover B, 
A including B, A comprising B, 
A embracing B, A containing B. 

AB. 
All A is all B, 

A = B, B = A. 
A alone is B alone, only A is only B, 
Sole A is sole B, entire A is entire B, 

A = B, B = A. 
Some not-A is some not-B, Certain not A's are some not- 
B's, one at least of the not-A's is not-B, Several not-A's 
are some not-B's, Most not-A's are some not-B's, Some- 
times not-A is some not-B. 
ab = ab. 



§ 474.] EXAMPLES. 285 

All A is some B, Every A is some B, 
Each A is some B, B only is A, B alone is A, B solely is 
A, Nothing but B is A, Nothing else than B is A: 
A = AB. 
No A is B, Not any A is B, A is never B, A is distinct 
from B, A always excludes B, 
No A = B. 
Provided not-A is not-B, Supposing not-A is not-B, Pre- 
suming not-A is not-B, Admitting not-A is not-B, 
Although not-A is not-B, Where not-A is not-B, When 
not-A is not-B, Wherever not-A is not-B: 
If a = b. 
A or B, not-A or not-B, 

Ab | aB. 
A or not-B: 

AB | ab. 
B or not-A: 

AB | aB. 
Unless A is B, A is C 

A if b, i. e., Ab = AbO. 
Some A is all B 

BA = B. 
Some B is all A 

AB = A. 
A'is not BG 

No A == BO. 
A is not either B or C 

No A = Be | bO. 
Except where, or, unless, 

I- 

A is in part B 

BA = B. 

A is partly not-B 

bA = bA. 



CHAPTER XIX. 



READING. 



475. In order to attain a proficiency in using the Reasoning 
Frame, it is desirable to learn to read it with ease. I will give 
some examples in reading for the purpose of assisting the rea- 
der to learn to read the various combinations in a Reasoning 
Frame. 

The readings are so numerous that I cannot begin to give all 
of them, but I will give those which I think are the most useful. 

These readings will also be t useful in furnishing formulae 
to the student when he comes to state and work out proposi- 
tions in the Reasoning Frame. 

476. Make an A diagram: 



A a 



Fig. 200. 



We can now read, 

(1) All A = All A 

(2) All a = All a 

(3) No a = A 

(4) Every A = A 

(5) No A = a 



§§ 477, 478.] EXAMPLES. 

477. Make an AB diagram : 



287 



A 


a 








B 






b 



Fig. 201. 

We can now read, 

(1) A = either B | b 

(2) b = A | a and conversely, 

(3) AB == AB 

(4) Not any A = a 

(5) Not any A = either aB | ab 

478. Make an AB diagram and eliminate the combination 
ab by making a figure 1 in it. 



A 


a 








B 




1 


b 



Fig. 202. 

We can now read, 

(1) No a = ab and conversely. 

(2) No b = ab and conversely. 

(3) Not any a = b and conversely. 

(4) B = A | a and conversely. 

(5) Only A = b and conversely. 

(6) B alone = a. 

(7) Every a = B and conversely. 

(8) Each b = A and conversely. 



288 



READING. 



LChap. 19. 



479. 



(9) If b =^= bA, then a = aB and conversely. 

(10) When b = bA, a = aB and conversely. 

(11) When a = aB, b = bA and conversely. 

(12) Provided a = aB, b = bA and conversely. 

(13) a = No b. 

(14) b = No a. 
Make an ABC diagram: 



AB 


Ab 


aB 


ab 




■• 




1 




C 








1 


c 



Fig. 203. 

Eliminate the combinations aBC, abc, by making a figure 1 
in those sections. 
We can now read, 

(1) All a = either bC | Be and conversely. 

(2) Not any a = either BC | be and conversely. 

(3) BC | be = ABC | Abc and conversely. 

(4) BC = BCA and conversely. 

(5) bC | Be = all a and conversely. 

(6) Not either BC | be = a and conversely. 
480. Make an ABCD diagram: 



AB 


Ab 


aB 


ab 












CD 


1 






1 


Cd 


1 






1 


cD 










cd 



Fig. 204. 



§§ 481, 482.] 



EXAMPLES. 



289 



Eliminate the combinations ABCd, ABcD, abCd, abcD, by 
making a figure 1 in those sections. 
We can now read, 

(1) AB = No Cd | cD. 

(2) Cd | cD = No AB. 

(3) All Cd | cD = Ab | aB. 

(4) No Cd | cD = AB | ab and conversely. 

(5) If A = B, C = D, c = d. 
481. Make an ABCD diagram: 



AB 


Ab 


aB 


ab 






1 




1 


CD 


1 




1 




Cd 








cD 










cd 



Fig. 205. 

Eliminate the combinations ABCd, AbCD, aBCd, abCD by 
making a figure 1 in those sections. 
We can now read, 

(1) All Bd | bD = Ac | aC. 

(2) No Bd | bD = AC | ac. 
482. Make an ABCD diagram: 



AB 


Ab 


aB 


ab 








1 




CD 








1 


Cd 


1 








cD 




1 






cd 



19 



Fig. 206. 



290 



READING. 



[Chap. 19. 



Eliminate the combinations ABcD, Abed, aBCD, abCd. 
We can now read, 

(1) All Ac | aC === Bd | bD. 

(2) No Ac | aO = BD | bd. 

(3) All BD | bd = AC | ac. 

(4) ab = No Cd. 
483. Make an ABCD diagram: 



AB 


Ab 


aB 


ab 




1 








CD 




1 






Cd 




1 






cD 




1 






cd 



Fig. 207. 

Eliminate tne combinations ABCD, AbCd, AbcD, Abed. 
We can now read, 

(1) No AB == CD, and conversely. 

(2) Never when A = B, C = D 

(3) A = B or CD = CD 

(4) If A = b, thenC = D 

484. Make an ABCD diagram: 



AB 


Ab 


aB 


ab 








1 


] 


CD 








Cd 




1 






cD 




1 


1 


1 


cd 



Fig. 208. 

Eliminate the combinations abCD, aBCd, AbcD, Abed, 
aBcd, abed. 



§§ 485, 486.] 



EXAMPLES. 



201 



We can now read, 

(1) a = D | b = C and conversely. 

(2) If c = d, then A = B 

485. Make an ABCD diagram: 



AB 


Ab 


aB 


ab 








1 




CD 


1 


1 





1 


Cd 




1 


cD 








1 


cd 



Fig. 209. 

Eliminate the combinations aBCD, ABCd, AbCd, abCd, 
abcD, abed. 

We can now read, 

(1) a = B | C = D and conversely. 

(2) If a = b, then C = D 

(3) If O =±= d, then a = B 

486. Make an ABCD diagram: 



AB 


Ab 


aB 


ab 








1 


1 


CD 










Cd 


1 




1 




cD 


1 






1 


cd 



Fig. 210. 

Eliminate the combinations aBCD, abCD, ABcD, aBcD, 
ABcd, abed. 
We can now read, 



READING. 



[ Chap. 19, 



(1) a = d | c = b and conversely. 

(2) ab = | D 

(3) CD = A 

(4) cD = Ab | ab 

(5) aB = Cd | cd 

(6) Cd == Cd 

(7) Ab f= Ab 

(8) Cd = A | a 

(9) Ab = C | c 

487. Make an ABCD diagram: 



AB 


Ab 


aB 


ab 




1 






1 


CD 




1 


1 




Cd 




1 


1 




cD 


1 






1 


cd 



Fig. 211. 



Eliminate the combinations 
aBCd, aBcD, abCD, abed. 
We can now read, 

(1) Ab | aB = CD 

(2) Cd | cD = AB 
Also the following pair: 

(3) Ac | aC = BD 

(4) Bd | bD = AC 
Also the following pair: 

(5) Be | bC = AD 

(6) Ad | aD = BC 
And the following pair: 

(7) ad | AD = cB | 

(8) cb | CB = aD | 



ABCD, ABcd, AbCd, AbcD, 



| cd and conversely, 
ab and conversely. 

bd and conversely. 
| ac and conversely 

ad and conversely, 
be and conversely. 

Cb and conversely. 
Ad and conversely. 



489, 490.] EXAMPLES. 

488. Make an ABCD diagram: 



293 



AB 


Ab 


aB 


ab 




1 








CD 




1 


] 


1 


Cd 




1 







cD 




1 




cd 



Fig. 212. 

Eliminate the combinations containing ABCD, AbCd, Abe, 
aCd. 

We can now read, 

(1) A = B | C = D and conversely. 

(2) a == aCD | ac 

(3) c = cAB | ca 

(4) If A = b, G = D 

489. Make an ABCD diagram: 



AB 


Ab 


aB 


ab 




1 






1 


CD 




1 


1 




Cd 


1 








cD 




1 


1 


1 


cd 



Fig. 213. 

Eliminate the combinations ABCD, ABcD, AbCd, Abed, 
aBCd, aBcd, abCD, abed. 

We can now read the following pair of propositions: 

(1) aD | Ad = cB | CB and conversely. 

(2) ad | AD = Cb | cb and conversely. 



294 READING. 

490. Make an ABGD diagram: 



[ Chap. 19. 



AB 


Ab 


aB 


ab 




1 




1 




CD 




1 




1 


Cd 


1 




1 




cD 




1 




1 


cd 



Fig. 214. 

Eliminate the combinations ABCD, ABeD, AbCd, Abed. 
aBCD, aBcD, abCd, abed. 

We can now read the following pair of propositions: 

(1) aD | AD = cb | Cb 

(2) cB | CB = Ad | ad 

491. Make an ABCD diagram : 



AB 


Ab 


aB 


ab 












CD 










Cd 






1 


1 


cD 






1 


1 


cd 



Fig. 215. 

Eliminate the ac combinations. 
We can now read, 

(1) Eveiwthing = A | aC 

(2) c = cA 

(3) a = aC 

(4) If a =* aC, then c == cA 



§§ 492, 493.] EXAMPLES. 

492. Make an ABCD diagram: 



295 



AB 


Ab 


aB 


ab 




1 








CD 


1 


] 


1 


Cd 








cD 
cd 









Fig. 216. 

Eliminate the combinations containing ABCD, AbCd, aCd. 
We can now read, 

AB = AB | C = D 
493. Make an ABCD diagram: 



AB 


Ab 


aB 


ab 




1 








CD 




1 






Cd 




1 






cD 




1 






cd 



Fig. 217. 

Eliminate the combinations containing ABCD, AbCd, Abe. 
We can now read, 

(1) A = B | CD = CD 

(2) Ab = CD 

(3) CD = Ab | a 



6 ' READING. 

494. Make an ABCDEF diagram: 



[ Chap, 19. 



ABO 


ABc 


AbC 


Abe 


aBC 


aBc 


abC 


abc 




1 
















DEF 






1 


1 


1 


1 


1 


1 


DEf 


















DeF 


















Def 


















dEF 


















dEf 


















deF 


















def 



Fig. 218. 



Eliminate the combinations containing ABCDEF, and all the 
combinations containing ABc, excepting the one containing 
ABcDEF, and all the combinations containing DEf, excepting 
the one containing ABODEf. 

The Reasoning Frame now shows the visible expression of 
the proposition, 

AB = C I DE = F 



495.J EXAMPLES. 

495. Make an ABCDEF diagram : 



297 



ABC 


ABc 


AbC 


Abe 


aBC 


aBc 


abC 


abc 




1 
















DEF 










1 


1 


• 


1 


DEf 










1 


1 


1 


1 


DeF 










1 


1 


1 


1 


Def 


















dEF 


















dEf 


















deF 


















def 



Fig. 219. 



Eliminate the combination containing ABCDEF, and also 
the combinations containing ABc, AbC, Abc, excepting those 
containing DEF; and also the combinations containing DEf, 
DeF, Def, excepting those containing ABC. 

The Reasoning Frame now shows us the visible expression 
of the proposition, 

A = BC I D = EF 



298 READING. 

496. Make an ABCDEF diagram : 



[ Chap. 19. 



ABC 


ABc 


AbC 


Abe 


aBC 


aBc 


abC 


abc 












1 




1 


1 


DEF 










1 




1 


1 


DEf 










1 




1 


1 


DeF 










1 




1 


1 


Def 


1 


1 


1 


1 


1 




1 


1 


dEF 












1 






dEf 


1 


1 


1 


1 


1 




1 


1 


deF 


1 


1 


1 


1 


1 




1 


1 


def 



Fig. 220. 



Eliminate the combination containing aBcdEf, and also the 
combinations containing aBC, abC, abc, excepting those con- 
taining dEf; and also the combinations containing dEF, deF, 
def, excepting those containing aBc. 

The Reasoning Frame now shows the visible expression of 
the proposition, 

a = Be I d = Ef 



498] EXAMPLES. 

497. Make an ABCDEF diagram: 



299 



ABC 


ABc 


AbC 


Abe 


aBC 


aBc 


abC 


abc 








1 


1 




1 


1 




DEF 
















1 


DEf 






1 






1 




1 


DeF 






1 












Def 






1 






1 




1 


dEF 






1 












dEf 






1 






1 




1 


deF 






1 












def 



Fig. 221. 

Eliminate the combination containing abcDEf ; and also the 
combinations containing AbC, Abc, abC, excepting those con- 
taining DEf; and also the combinations containing aBc, except 
ing those containing f ; and also the combinations containing 
abcDeF, abcdEF, abcdeF. 

The Reasoning Frame now shows the visible expression of 

the proposition, 

b = DE | ac = f 

498. Make an ABCD diagram: 



AB 


Ab 


aB 


ab 




1 
1 


1 






CD 


1 




Cd 
cD 




1 


1 




1 


1 


cd 



Fig. 222. 



800 



READING. 



[Chap. 19. 



Eliminate the combinations containing ABO, Ab, aBc. 
We can now read, 

A = B | B = C and conversely. 
499. Make an ABCD diagram: 



AB 


Ab 


aB 


ab 






1 


1 


1 


CD 


1 






Cd 


1 


cD 




1 




cd 



Fig. 223. 

Eliminate the combinations containing ABCd, AbCD, Abe, 
aCD. 

We can now read, 

(1) If A == b, === d J A — B 

(2) If C = D, A = B | C = d 

(3) A = B | C = d 

500. Let us take this example: 

a | D = B | C 

The easiest method of finding the combinations which a 
proposition in this form will cause us to eliminate, is to first 
make an ABCD diagram, and letter it according to the circum- 
stances of the case, remembering that a letter and its oppo- 
site cannot occupy the same section; eliminate the combina- 
tions which are inconsistent, then make another ABCD dia- 
gram. Letter it in the usual manner and transfer to it the 
eliminated combinations. 

The given proposition can be stated thus: 

(1) ad | AD = Be | Cb 



§ 500 .j LETTERING DIAGRAMS. 

Make an ABCD diagram: 



301 



aD 


ad 


AD 


Ad 






1 


1 




BC 










Be 










bC 




1 


1 




be 



Fig. 224. 

Now, if a | D = B | C, then the combinations containing 
aBCd, abed, ABOD, AbcD, are inconsistent, and we eliminate 
them by making a figure 1 in those sections. 

Make another ABCD diagram: 



AB 


Ab 


aB 


ab 




1 








CD 






1 




Cd 




1 






cD 








1 


cd 



Fig. 225. 

Eliminate the combinations aBCd, abed, ABCD, AbcD. 

The Reasoning Frame now shows the visible expression of 
the proposition, 

a | D = B | C 

This method of lettering a Reasoning Frame, according to 
the exigencies of the occasion, in order to learn the combina- 
tions which are to be eliminated, is of frequent usefulness. 



CHAPTER XX. 

THE SYLLOGISM. 

501. Thus far we have not discussed the Syllogism, because 
in our system, we have no use for it. But as a work on Logic 
without it would not be considered complete by the majority 
of educated people, we will give some space to this system of 
reasoning. 

The Syllogism is the principal method of deductive infer- 
ence, that is, an inference from the general to the particular, 
employed by the old logic. It should contain just three terms, 
a subject and predicate of the conclusion; another term called 
the middle term, which occurs in both premises. 

502. It has three propositions, viz.: two premises and a 
conclusion. The premise containing the major term is called 
the major premise; the major term is the predicate of the con- 
clusion. The premise containing the minor term is called the 
minor premise; the minor term is the subject of the conclusion. 

503. Syllogisms are divided into four figures. The position 
of the middle term determines the figure. The middle term 
is the term which occurs in both premises and not in the con- 
clusion. 

504. In the first figure the middle term is subject in the 
major premise and predicate in the minor premise, thus: 

B = 

A = B 

Therefore, A = C 

505. In the second figure the middle term is predicate in 
both premises, thus: 

C = B 
A = B 

506. In the third figure the middle term is subject in both 
premises, thus: 



§§ 507-515.] FIGURES. 303 

B = C 
B = A 

507. In the fourth figure the middle term is predicate in the 
major and subject in the minor premise, thus; 

C = B 
B == A 

508. The propositions in a syllogism are divided into four 
forms. I think that E is the only definite proposition in these 
four forms. A, I and O are indefinite and must be converted 
into definite propositions before we can use them. Until they 
are thus converted they should have no place in an exact logic. 

509. The first is the Universal Affirmative: 

"All men are animals/' A = AB, 
and its symbol is A. 

510. The second is the Universal Negative: 

"No men are infallible/' No A = B, 
and its symbol is E. 

511. The third is the Particular Affirmative: 

"Some men are wise/' AB = AB, 
and its symbol is I. 

512. The fourth is the Particular Negative: 

"Some men are not religious/' Ab = Ab, 
and its symbol is O. 

513. The symbols A and I are taken from the latin word 
affirmo, "I affirm/' and E and O are taken from the latin word 
nego, "I deny." 

514. The first figure only, will yield conclusions in all the 
forms A, E, I, O. The second figure yields negative conclu- 
sions. The third figure yields particular conclusions. The 
fourth figure does not yield Universal Affirmative Conclusions. 

515. The order of subject and predicate varies in the minds 
of persons according to the idea which they wish to convey. 

"The best form of government is government by a plurality 
of persons/' and 

"Government by a plurality of persons is the best form of 
government/' 



B04 SYLLOGISM. [ Chap. 20. 

would be stated in different figures, although both proposi- 
tions have substantially the same meaning. 

516. When the middle term embraces both the major and 
the minor terms, it naturally forms the predicate of both pre- 
mises. This makes the second figure. 

517. When the middle term is smaller than the major 
and minor terms it naturally forms the subject of both pre- 
mises. This makes the third figure. 

518. The syllogism is governed by six rules: 

(1) Every syllogism must have three and only three 
terms. 

(2) There must be three and only three propositions. 

(3) The middle term must be distributed (that is, taken 
altogether), at least once in the premises. 

(4) No term undistributed (i. e., taken partially) in the 
premises, must be distributed in the conclusion. 

(5) There can be no conclusion drawn from negative 
premises. 

(6) If one premise be negative, the conclusion must be 
negative. 

519. The premises are so termed because they premise or go 
before the conclusion. 

The conclusion is so named because it concludes, or shuts 
up in one the major and minor propositions. 

520. In this syllogism, 

"All horned animals ruminate" (major proposition). 
"A sheep is a horned animal" (minor proposition). 
Therefore, "A sheep ruminates" (conclusion). 
"Sheep" is named the minor term because it is less extensive 
than "ruminating." 

"Ruminate" is the major term, because it includes "sheep." 
"Horned animals" is the middle or mean term. 

521. The rule or maxim which is commonly called dictum 
de omni et nidlo, by which Aristotle explains the validity of a 
syllogism in this form, 

Every B is C 
Every A is B 



§§ 522, 523.] RULES. 805 

Therefore, every A is C, 
is this: Whatever is predicated of a distributed term, whether 
affirmatively or negatively, may be predicated in like manner 
of everything embraced by it. This maxim, however, cannot 
be applied to all syllogisms. For instance, the dictum can- 
not be applied to this valid syllogism: 

No savages have the use of metals, 

The ancient Germans had the use of metals, 

Therefore, they were not savages. 

522. There are more rules which are used to test the valid- 
ity of syllogisms: 

1st. If two terms agree with one and the same third, 

they agree with each other. 
2d. If one term agrees and another disagrees with one 
and the same third term, these two disagree with each 
other. 
The first of these rules tests the validity of affirmative con- 
clusions; the second, of negative conclusions. 

523. Let us take this example and work it out by our 
system : 

(1) All horned animals ruminate, 

(2) A sheep is a horned animal, 

(3) Therefore a sheep ruminates. 
Let A = sheep, 

B = horned animal, 
C = ruminates. 
The propositions can be stated thus: 

(1) B = BC 

(2) A == AB 

(3) Therefore, A = AC 

The term "ruminate" is a wider term than the term "horned 

animal," and in order that our proposition may be strictly 

accurate, we must reduce the term "ruminate" to an equality 

with the term "horned animal." We do this by adding the 

term "horned animals" to the term "ruminate," so that the 

term "horned animals ruminating," is the equivalent of the 

term "horned animal." 
20 



306 



SYLLOGISM. 



[ Chap. 20. 



524. Where the predicate is larger than the subject, we 
reduce the predicate to the limits of the subject by adding the 
subject to the predicate. It is a kind of subtraction. By add- 
ing B, which stands for "horned animal," to C, which stands 
for "ruminating animals," we, in effect, subtract from "rumi- 
nating animals" all the not-horned animals, and this reduces 
the "ruminating animals" to the "horned animals." We thus 
have this paradox, that by adding a limited term to a more 
extensive term, we take away from the meaning of the exten- 
sive term. By adding a term we subtract from the meaning, 
except where the terms are equivalents. 

To be formally accurate, we must always state our propo- 
sitions so that they will be true when read either way. It can 
always be done by adding the subject to the predicate, where 
the predicate is larger than the subject. 

Now, if B = BC, then the combinations containing Be are 
inconsistent, because they imply that B = c, and we there- 
fore eliminate them by making a figure 1 in those sections 
of an ABC Reasoning Frame: 



AB 


Ab 


aB 


ab 






2 






C 


1 


2 


1 




c 



Fig. 226. 



In our system, propositions in the form of B = BO, i. e., 
propositions where the subject is added to the predicate, are 
never worked backward. 

Now, if A = AB, then the combinations containing Ab are 
inconsistent, and we eliminate them by making a figure 2 in 
those sections. 

In examining the Reasoning Frame, we find that there is 
only one section containing the letter A, so that we can define 



§ 524.] AN EXAMPLE. 307 

A by the other letters in that section; they are B and C. But 
we cannot say that A = BC, because that implies that BC is A; 
there is another BC which prevents us from saying that BO 

The definition of A is, A = ABC, but in translating our 
terms we need not repeat the subject, thus we can read, 
A = ABC, which we can translate, 

A sheep is a horned animal and is a ruminating animal. 

Now the syllogism drops a part of this information and 
simply gives us, . 

A sheep is a ruminating animal. 

This is allowable because we are not obliged to give any 
more information than is necessary to serve the purpose in 
view. 

When a letter is found in more sections than one, its defini- 
tion will be in the form of a disjunctive proposition. 

Now, the term B occurs in two combinations, viz.: ABC 
and aBC. We cannot say, when a term occurs in two combi- 
nations, that it is either one of them alone; our definition must 
be that it is one or the other; hence 

B = BAC or BaC, which we can translate, 

A horned animal is a sheep and a ruminating animal, or it is 
not a sheep but is a ruminating animal. 

Now, where we have a term and its negative, like A and a, 
occurring in two alternants, we can omit translating them, 
and we can contract the definition just given into, 
B = BC, which we can translate: 

A horned animal is a ruminating animal. 

Again, the letter C occurs in three combinations, viz. : ABC, 
aBC and abC, so that the definition of C is: 
C = AB | aB | Cab 

The reason why we do not repeat C before AB and aB is 
because AB = C and aB = C; but we repeat C before ab 
because there is another combination containing ab, so that 
we could not say that ab = C. 

Now we can translate the definition of C thus: 
A ruminating animal is either, 



308 SYLLOGISM. [ Chap. 20. 

A sheep and a horned animal; or 
Not a sheep but a horned animal ; or 
Not a sheep and not a horned animal. 
The definition of a is, 

a = aBC | abC | abe, which can be translated: 
An animal which is not a sheep is either a horned animal 

and a ruminating animal ; or 
Not a horned animal but a ruminating animal; or 
Neither. 
When the definition of b is, • 

b = baC | bac, then, 
in this case we can omit the and c and translate the definition 
of b, thus: 

What is not a horned animal is not a sheep. 
The definition of c is, 

c = cab, which we can translate: 
What is not a ruminating animal is neither a sheep nor 
a horned animal. 

525. The reader can now see by this example, that the 
syllogism gives us a part of the information contained in two 
propositions having a common term. The reasoning process 
is a process of finding all the equivalents, inferences, consis- 
tents, inconsistents, and contradictories of gflven propositions. 
It is also a process of conversion ; tha>t is, by the reasoning pro- 
cess we can convert given propositions into many other equiva- 
lent forms. But the syllogism is a very imperfect instrument 
for effecting the possible conversions. 

526. Let us take this example: 

(1) No savages have the use of metals, 

(2) The ancient Germans had the use of metals, 

(3) Therefore they were not savages. 
Let A == savages, 

B == the use of metals, 
= the ancient Germans. 
Our propositions can be stated thus: 
No A = B 
C == CB 



§ 5-26.] 



AN EXAMPLE. 



309 



Therefore, C = Ca. 
We state the first proposition in the form of 
Xo A = B 

It is evident that the proposition will read backward thus: 
Those who have the use of metals are no savages. 

:. ;; () a = B, then the combinations containing AB are 

in< stent, and we eliminate them by making a figure 1 in 

those sections of an ABC Reasoning Frame: 



AB 


Ab 


aB 


ab 




1 


2 




2 


C 


1 








c 



Fig. 227. t 

Again, if C = CB, then the combinations containing Cb are 
inconsistent, and we eliminate them by making a figure 2 in 
those sections. 

An examination of the Reasoning Frame shows us that the 
definition of C is CaB, which we can translate: 

The ancient Germans had the use of metals and were 
not savages. 
The definition of A is, 

A = Abe, which we can translate, 
Savages do not have the use of metals, and are not 
ancient Germans. 
The definition of B is, 

B = BaC | Bac, which we can translate, 
Those who had the use of metals were not savages. 
The definition of a is, 

a = aBC | aBc, which we can translate, 
Those who are not savages, have the use of metals. 
The definition of b is, 



310 



SYLLOGISM. 



[Chap. ,20. 



b = bcA | bca, which we can translate, 
Those who do not have the use of metals are not the 
ancient Germans. 
The definition of c is, 

c = cAb | caB | cab, which we can translate, 
Those who were not ancient Germans were either sav- 
ages without the use of metals; or 
not savages without the use of metals; or 
neither. 
527. Let us take this example: 

(1) Some Europeans are Englishmen, 

(2) Some Englishmen are Londoners. 
Let A = Europeans, 

B = Englishmen, 
C = Londoners. 
The premises can be stated thus: 
(1) AB = B 
# (2) BC = G 

Make an ABC diagram: 



AB 


Ab 


aB 


ab 






2 


1 


2 


C 






1 




c 



Fig. 228. 



Now, if B = AB, then the combinations containing Ba are 
inconsistent, and we eliminate them by making a figure 1 in 
those sections. 

Again, if C = BC. then the combinations containing bC are 
inconsistent, and we eliminate them by making a figure 2 in 
those sections. 

From the uneliminated combinations we can get these defi- 
nitions: 



§528.] AN EXAMPLE. 311 

(1) C = CAB, which can be translated, 
Londoners are Europeans and Englishmen. 

(2) Ab == Abe, which can be translated, 
Europeans who are not Englishmen are not Londoners. 

(3) a = abc, which can be translated, 

Those who are not Europeans are not Englishmen and 
not Londoners. 

(4) b = be, which can be translated, 
Those who are not Englishmen are not Londoners. 

(5) c = cAB | cAb | cab, which can be translated, 
Those who are not Londoners are either Europeans and 

Englishmen, or Europeans who are not Englishmen, 
or neither Englishmen nor Europeans. 

(6) A = AG | Ac, which can be translated, 
Europeans are either Londoners or not Londoners. 

(7) B = BC | Be, which can be translated, 
Englishmen are either Londoners or not Londoners. 

We can also read, 

(8) No C = b, tfhich can be translated, 
No Londoners are not Englishmen. 

(9) No B = a, which can be translated, 
No Englishmen are not Europeans. 

528. Let us take this example: 

(1) Cornishmen are Englishmen, 

(2) Some Englishmen are Londoners, 
Let A = Cornishmen, 

B = Englishmen, 
C = Londoners. 

The premises can be stated thus: 

(1) A = AB 

(2) BC = C 

Make an ABC diagram: 



312 



SYLLOGISM. 



[ Chap. 20. 



AB 


Ab 


aB 


ab 






12 




2 


C 




1 




c 



Fig. 229. 

Now, if A = AB, then the combinations containing Ab are 
inconsistent, and we eliminate them by making a figure 1 in 
those sections. 

Again, if = BC, then the combinations containing bC are 
inconsistent, and we eliminate them by making a figure 2 in 
those sections. 

We can now read, 

(1) b = be, which can be translated, 

Those who are not Englishmen are not Londoners. 

(2) C = CB, which can be translated, 
Londoners are Englishmen. 

(3) No A = b, which can be translated, 
No Cornishmen are not Englishmen. 

(4) No C = b, which can be translated, 
No Londoners are not Englishmen. 

529. Speaking of the syllogism, Prof. Bain, in his "Deduc- 
tive and Inductive Logic," p. 207, says: 

"1. It is the peculiarity of the syllogism that the conclu- 
sion does not advance beyond the premises. This circum- 
stance has been viewed in two lights. On the one hand it is 
regarded as the characteristic excellence of the syllogism. 
On the other hand it is represented as constituting a petitio 
principii. 

In the syllogism, 

Men are mortal, 
Kiugs are men, 
Kings are mortal, 



§3 530-532] CRITICISMS. 313 

the conclusion seems already affirmed in the premises." 

530. For myself, I regard the syllogism as useless, but I do 
not think that the syllogism can be criticised for "not advanc- 
ing beyond the premises." It is true of all reasoning that it 
can ''not advance beyond the premises." It can turn the prem- 
ises into other forms; it can find many equivalents for the 
premises, but it cannot 'advance beyond" them. 

53.1. Prof. Bain again says: 

2. "There remains a far more serious charge and one that 
takes us direct to the root of formal reasoning. Supposing 
there were any doubt as to the conclusion that 'Kings are mor- 
tal,' by what right do we proclaim, in the major, that 'All 
men are mortal,' Kings included? 

It would be requisite seemingly, to establish the conclusion 
before we can establish the major. 

In order to say, 'All men are mortal,' we must have found, 
in some other way, that all kings and all people are mortal. So 
that the conclusion first contributes its quota to the major 
premise, and then it takes it back again. 

This is the dead-lock of the syllogism, the circumstance 
that has brought down upon it the charge of 'reasoning in a 
circle' (petitio principii). In point of fact we can hardly pro- 
duce a more glaring case of that fallacy." 

532. Neither do I consider this a serious charge against 
the syllogism. When the proposition, "All men are mortal" 
is given, logic does not question whether that is true or not. 
It is not a question of logic, it is a question of fact. Logic 
takes it for granted, assumes it to be true, and proceeds to 
find its various equivalents. There is no fallacy in the argu- 
ment. A syllogism may be perfectly valid though the premises 
are false. We can draw the conclusions from false prem- 
ises and the operation be logically valid. Logic has no power 
to pass on the truth or falsity of the premises; that is outside 
of its domain; its function is to draw the proper conclusions, 
that is, the conclusions which necessarily follow from the prem- 
ises given. 



314 SYLLOGISM. [Chap. 20. 

533. Miss Jones in Elements of Logic, p. 161, makes a just 
criticism against the ordinarily accepted rules of the syl- 
logism. 

Speaking of terms and term-names she says: 

"For if, in e. g., the syllogism, 
All N's are Q's, 
Some R's are N's, 
Therefore, some R's are Q's. 

We call (1) all N's, (2) some Q's, (3) some R's, (4) some N's, 
Terms, then the rule that in a valid syllogism we must have 
only three terms, excludes all syllogisms except those in Fig. 3, 
which have the middle term distributed twice. In the instance 
above taken we have four terms." 

534. I think Miss Jones is correct in the position taken that 
all N's and some N's are two different terms. "All" and 
"some" are not equivalents; they refer to two different groups 
of things, and each group can be expressed logically by a dif- 
ferent symbol, and this will give us four terms. 

535. Lotze in his work on Logic, p. 114, says: 
"Following Aristotle we give the name of Inference or Syl- 
logism to any combination of two judgments for the produc- 
tion of a third and valid judgment which is not merely the sum 
of the two first. Such production would be impossible if the con- 
tents of the antecedent judgments, the two premises were entirely 
different; it is only possible if they both contain a common ele- 
ment M, the middle concept or terminus medius, which the 
one relates to S, the other to P (here M stands for middle term, 
S stands for subject, P stands for predicate)." 

536. We can combine by our system two entirely different 
premises. 

Let us take these propositions: 

(1) Washington is the capital of the United States. 

(2) Salt is chloride of sodium. 
Let A = Washington 

B = Capital of the United States. 
C = Salt, 



§536] 



INDEPENDENT PROPOSITIONS. 



315 



D = chloride of sodium. 
The propositions can be stated thus: 

(1) A = B 

(2) B = A 

(3) C = D 

(4) D = C 

The question is: What inferences can we draw from these 
propositions? 

Now, if A = B then the combinations containing Ab are 
inconsistent, and we eliminate them by making a figure 1 in 
those sections of an ABCD Reasoning Frame: 



AB 


Ab 


aB 
2 


ab 


CD 




1 




3 
4 


13 
41 


23 
24 


3 


Cd 

cD 

cd 


4 


1 


<? 





Fig. 230. 

Again, if B = A, then the combinations Ba are inconsistent 
and we eliminate them by making a figure 2 in those sections. 

Again, if C = D, then the combinations containing Cd are 
inconsistent and we eliminate them by making a figure 3 in 
those sections. 

Again, if D = 0, then the combinations containing Dc are 
inconsistent and we eliminate them by making a figure 4 in 
those sections. 

From the combinations which remain we can get the fol- 
lowing definitions: 

(1) AB 1 ab = CD | cd 

(2) AD | ad = BC | be 

(3) AC I ac == BD ! bd, 
Which we can translate as follows: 

(1) Either Washington and the capital of the United States, 



316 SYLLOGISM. [Chap 20. 

or neither, is salt and chloride of sodium or neither. 

(2) Either Washington and chloride of sodium, or neither, 
is the capital of the United States and salt, or neither. 

(3) Either Washington and salt, or neither, is the capital of 
the United States and chloride of sodium, or neither. 

There are other definitions which can be obtained such as, 
AB = ABGD | ABcd, which can be translated, 

Washington and the capital of the United States are 
salt and chloride of sodium or neither. 
537. I give this example merely to show that it is possible 
to draw conclusions from entirely different premises. The 
usual way of obtaining the readings by our system, is to 
read the combinations which remain uneliminated in the Reas- 
oning Frame, but as heretofore explained, we can also obtain 
readings by prefixing the word "No" to the eliminated com- 
binations in which two or more letters are eliminated. Thus, 
we can take the example just given and get these conclusions 
by prefixing the word "No" to these eliminated combinations : 

(1) No A == Ab, 
which can be translated, 

No Washington is not the capital of the United 
States. 

(2) No b = bA, 
which can be translated, 

No not-capital of the United States is Washington. 

(3) No a = aB, 
which can be translated, 

No not- Washington is the capital of the United States. 

(4) No B = Ba, 
which can be translated, 

No capital of the United States is not- Washington. 

(5) No C = Cd, 
which can be translated, 

No salt is not chloride of sodium. 
((>) No d == Cd, 
which can be translated, 



§ 538 ] VENN ON THE SYLLOGISM. 317 

No not-chloride of sodium is salt. 

(7) No c = cD, 
which can be translated, 

No not-salt is chloride of sodium. 

(8) No D = Dc, 
which can be translated, 

No chloride of sodium is not-salt. 
We could also get a good many more readings of this kind, 
such as: 

No AB = Cd, etc., etc. 
We could till a chapter with the various combinations that 
can be read in this way from the eliminated combinations in 
this example, but it is hardly necessary to do so. 

538. Prof. Venn in his Symbolic Logic, p. 402, makes the 
following very sensible remarks on the syllogism : 

"We must frankly remark that from our point of view we 
do not greatly care for this venerable structure, highly useful 
though it be for purposes of elementary training in thought 
and expression, and almost perfect as it is technically, when 
regarded from its own standing point. But its ways of think- 
ing are not ours, and it obeys rules to which we own no allegi- 
ance. To it the distinction between subject and predicate is 
essential; to us this is about as important as the difference 
between the two ends of a ruler which one may hold either 
way at will. To it the position of the middle term is conse- 
quently worth founding a distinction upon. To us this is as 
insignificant as is the order in which one adds up the figures 
in an addition sum. On the other hand, the distinction 
between Universal and Particular propositions, which to it is 
vital, is to us unimportant. There are reasons, nevertheless, 
for taking some account of the syllogism here; partly because 
the contrast of treatment will serve to emphasize this differ- 
ence in the point of view, x>artly because the omission of any 
such references might possibly be taken as a confession of fail- 
ure on the part of the Symbolic logic. 

"Since the syllogism is a sound process, it must admit of some 
kind of treatment upon any scheme. There are two ways of 



318 SYLLOGISM. [ Chap. 20. 

treating it. The method which would naturally be adopted by 
any one familiar with the use of symbols, but entirely ignorant 
of logical tradition, would probably be this. He would begin 
by rejecting all distinction of figure as utterly alien to his 
scheme; and as the common system admits that the other 
three figures can be reduced to the first, he would insist upon 
this simplification being made before he took the work in hand. 
That is, he would take account only of the first four moods. 
Then he would go on to reduce these by the consideration that, 
to his thinking, X and not-X being both classes of the same 
essential character, there was no occasion to formulate a dis- 
tinction between moods which involved a negative and those 
which contained only affirmative premises. There would then 
remain only the distinction between a form which draws a uni- 
versal and one which draws a particular conclusion." 

539. Prof. Venn is a disciple of Boole and his work on Sym- 
bolic Logic is on the Booleian system, which is a sort of alge- 
braical logic. The reader can see that in our system the dis- 
tinction between subject and predicate is unessential ; neither 
do we care anything about a middle term. The distinctions 
of figure and mood are useless to us, and we can treat nega- 
tive and affirmative propositions with equal facility. 

540. So far as the theory of logic is concerned, there is a 
strong analogy in many points between the Booleian system 
and ours. 

541. Lotze in his work on Logic, vol. 2, p. 1, speaking of the 
syllogism, justly remarks: 

"True conclusions, as Aristotle has observed, can be cor- 
rectly drawn from false premises." 

Every Laplander is a born poet, 
Homer was a Laplander, 
And therefore, by the first figure a poet. 
All parasitic plants have red flowers, 
No rose has red flowers, 

Therefore, by the second figure roses are not para- 
sitic plants. 



§541.] LOTZE'S VIEW. 319 

Metals do not conduct electricity, 
All metals are non-fusible, 

And hence, according to the third figure, non-fu- 
sible substances exist which are non-conductors 
of electricity. 
Alter Laplander into Greek, plants which have red flowers 
into plants which have exploding seed vessels, and write glass 
for metal, and in each example one premise will be true, while 
by inserting a new middle term in each case, you may make 
both premises true, but in every case the conclusion follows 
with neither more nor less validity. 

Let T be a perfectly true proposition, S its subject and P its 
predicate; then a middle term, M, may be chosen at random, so 
long as the terms are arranged in both premises on the model 
of an Aristotelian figure. If this is done, the conclusion T will 
always follow according to the figure. 

We shall see why this is universally true if we take as our 
middle term an abstract symbol, M, instead of a concrete term, 

thus: 

All M are poets 

Homer was an M, 

All parasitic plants are M 

Roses are not M, 

All M are non-conductors 

All M are non-fusible. 
What these symbolic premises tell us, is the relations in 
which S and P must stand to some middle term, if their con- 
junction, SP, is to be valid in the conclusion; and, conversely, 
these premises tell us that given any middle term, M, to which 
S and P are related as required, then the proposition SP must 
be valid. If the M is found, and so both the required premises 
established, then SP is valid, not merely in fact, but now also 
of necessity; on the other hand, if we could show that there 
exists no M to which S and P can stand in the requisite relation 
we shall know that SP was impossible, for no experience could 
give us SP as a fact; but if we have merely chosen a wrong M, 
then the case is different. The premises we have chosen will 



320 SYLLOGISM. [Chap 20. 

not do, but there is" no reason why there should not be some 
other M, the insertion of which will render the premises correct 
and so necessitate the conclusion SP. 

If, again, we have correctly drawn a conclusion, SP, and that 
conclusion is unsound, there must be something false in the 
premises from which it follows. In a word, all cases where T 
is not given in direct perception, but deduced from premises, 
what really depends on the correctness of those premises is not 
the truth of T, but only our insight into that truth. Without 
correct premises, T cannot be true, but nevertheless it can be 
proved, and its truth is independent of any errors we may com- 
mit, when reflecting about it, and subsists even when conclu- 
sively deduced from premises materially false. This point 
deserves notice, for it is a common mistake in reasoning, to take 
the invalidity of the truth which is offered for T, as a proof of 
the falsehood of T itself, and to confuse the refutation of an 
argument with the disproof of a fact." 

542. Let us take this example of a pretended syllogism: 
Some men are kings 
Ail cooking animals are men 
Therefore all cooking animals are kings. 
This example is said to exemplify the fallacy of undistrib- 
uted middle. The middle term is "men," and it is not distrib- 
uted, that is, taken altogether, in either the major or the minor 
premise. 

The proposition 

All cooking animals are men, means, 
All cooking animals are some men, 
but this group of "some men" is a different group from the 
group in "some men are kings." 
Let A = men 
B = kings 
C = cooking animals. 
The premises can be stated thus: 
(1) BA = B 
' (2) C = CA 



543.] 



A FALLACY. 
Make an ABC diagram: 



321 



AB 


Ab 


aB 


ab 








12 


2 


C 






1 




c 



Fig. 231. 

Now, if B = BA, then the combinations containing aB are 
inconsistent, and we eliminate them by making a figure 1 in 
those sections. 

Again, if C = CA, then the combinations containing aC are 
inconsistent, and we eliminate them by making a figure 2 in 
those sections. 

From the uneliminated combinations we can get these defini- 
tions: 

(1) C = CAB | CAb 
which can be translated, 

Cooking animals are* men, and either kings or not kings. 

(2) a = abc, 
which can be translated, 

What are not men are neither kings nor cooking ani- 
mals. 
From the eliminated combinations we can get these defini- 
tions: 

(1) No B = a, 
which can be translated, 

No kings are not-men. 

(2) No C = a, 
which can be translated, 

No cooking animals are not-men. 

543. Frof. Bain remarks, p. 149: "We may have premises 

free from the last-named vice of undistributed middle, yet 

made to yield a false conclusion by overstepping the present 

rule (no term undistributed in the premises must be distributed 

21 



322 



SYLLOGISM. 



[ Chap, 20. 



in the conclusion) or raising a term of particular quantity, in 
the premises, to the rank of universal quantity in the conclusion. 
To this error is given the name Illicit Process; and according as 
the unduly extended term occurs in the major or minor premise, 
the error is called Illicit Process of the Major, or Illicit Process 
of the Minor. 

544. Let us take this example of the Illicit Process of the 
Minor: 

All men are mortal 

Some extended things are men 

Therefore, all extended things are mortal 
Let A = men 

B = mortal 

C = extended things. 
The premises can be stated thus : 

(1) A == AB 

(2) CA = A 
Make an ABC diagram: 



AB 


Ab 


aB 


ab 






1 






C 


2 


1 


2 




c 



Fig. 232. 

Now, if A = B, then the combinations containing Ab are 
inconsistent, and we eliminate them by making a figure 1 in 
those sections. 

CA = A will cause us to eliminate the combinations Ac. 

We eliminate them by making a figure 2 in those sections. 

From the uneliminated combinations we can get this defini- 
tion of C : 

C == CAB | CaB | Cab, 
which can be translated, 



§S 545, 546.] 



ILLICIT PROCESS. 



323 



Extended things are either mortal men, or mortals and 
not-ruen, or neither men nor mortal. 
545. Let us take the following example of Illicit Process of 
the Major: 

All men are fallible 

Some beings are not men 

Therefore, no beings are fallible. 
Let A = men 

B = fallible 

C = beings. 
The premises can be stated thus : 

(1) A = AB 

(2) Ca = Ca 
Make an ABC diagram : 



AB 


Ab 


aB 


ab 






1 






C 




1 






c 



Fig. 233. 

Now, if A = AB, then the combinations containing Ab are 
inconsistent, and we eliminate them by making a figure 1 in 
those sections. 

Ca = Ca will not cause us to eliminate any combinations. 
From the uneliminated combinations we can get this defini- 
tion of C: 

C = CAB | CaB | Cab, 
which can be translated, 

Beings are either fallible men, or fallible not-men, or 
neither fallible nor men. 
546. Prof. Bain, in "Deductive and Inductive Logic," p. 150, 
says: 

"There can be no conclusion drawn from negative premises. 
No men are gods 



324 



SYLLOGISM. 



[ Chap. 20. 



No trees are men 
do not supply the materials for a deductive inference. The 
reason of this is already apparent from what has been said as 
to the applying proposition, which must always affirm. To 
know only that two things are each excluded from a third 
thing, is to know nothing concerning their mutual relation." 
Let us work out the example given : 
Let A = men 
B == gods 
C = trees. 
The propositions can be stated: 

(1) No A = B 

(2) No C = A 

Now, if No A == B, then the combinations containing AB 
are inconsistent, and we eliminate them by making a figure 1 in 
those sections of an ABC Seasoning Frame. 

Make an ABC diagram: 



AB 


Ab 


aR 


ab 




12 


2 






C 


1 








c 



Fig. 234. 

Again, if No C = A, then the combinations containing AC 
are inconsistent, and we eliminate them by making a figure 2 in 
those sections. From the combinations which remain we can 
get the following definitions: 

A = Abe 
which can be translated, 

Men are not Gods and not trees. 
C = Ca 
which can be translated, 
Trees are not men. 



§§ 5i7 548.] MISS JONES' VIEW. 325 

B = aB 
which can be translated, 
Gods are not men. 
The reader can see that it is as easy to draw conclusions from 
negative premises as it is to do so from affirmative premises. 

547. We have already seen that by our system we can get 
all the conclusions w T hich are contained in the premises. 

Lotze justly criticises the barrenness of the syllogism. He 
says (p. 141) : 

"If we argue, 'Heat expands all bodies; iron is a body, there- 
fore, heat expands iron,' or 
All men are mortal 
Caius is a man 
Therefore, Caius is mortal, 
everyone will feel the barrenness of this procedure, and will 
reply, 'Undoubtedly heat expands all bodies, but each body in a 
different degree. Undoubtedly all men die, but the liability to 
die in one man is different from that in another. What we want 
to know for technical purposes or for administering a life 
insurance company is, how iron expands in distinction from 
lead, or how the mortality of Caius is to be estimated in distinc- 
tion from other men.' What good is it to say, 'If a man is 
offended, he gets angry; Caius is a man, therefore, if he is 
offended he will get angry?' " 

Now in our system we can state all the technical and scien- 
tific facts known in regard to the expansion of bodies by heat 
or the mortality of men, and in one operation draw all the 
conclusions which the facts would warrant. 

548. Miss Jones, in "Elements of Logic," p. 156, in criti- 
icsm of the syllogism, says: 

"It seems to me that Mill's criticism of the dictum cle omni ct 
nulla is well founded, and that the canon merely amounts to 
saying that that which is predicated of every member or every 
portion of a class, may be predicated of any member or any 
portion of that class. For, when we say, 'Whatever is predi- 
cated of a term distributed, may be predicated in like manner 



326 SYLLOGISM. [ Chap. 20. 

of everything contained under it/ it seems clear that by that 
term is meant the term name, for in, e. g. ? 

Some men are mortals, 
the subject "some men" is as much or as little (distributed) as 
in, 

All men are mortals, 
i. e. some class-name; and what is predicated of a class-name 
distributed is ex vi termini, predicated of each member of the 
class. 

The dictum is not a canon of syllogism, if syllogism means 
formal mediate inference, nor even a statement of relations 
between different classes, but merely a formulation of the 
truth that if any object or objects belong to a class what can 
be said about the class distributively, can be said about it 
or them. 

It seems to apply only in cases in which we are dealing with 
A, E, I, or O, propositions." 

549. Miss Jones' position in the above quotation that "some 
men is as much distributed as all men" is correct. "Some 
men" constitute a group of beings just as much as "all men" 
do, and "some men" is to be taken altogether just as much as 
"all men." 

One group we describe by "all," the other group we describe 
by "some." Logically the one group is as certain and definite 
as the other, and, as Miss Jones says, the one is as much or as 
little distributed as the other. 

550. She further says: "But a canon of syllogism ought to 
apply, whatever terms and term-names we are dealing with, 
and whatever admissible arrangement of these we are con- 
sidering. It ought to apply to syllogisms in the second, third 
and fourth figure, and to arguments in which all the terms are 
partial or single, as well as to syllogisms in figure 1, which 
have a class-name distributed for the subject name in the 
major premise. 

When Prof. Jevons says (Principles of Science, p. 9, 3d 
Ed.), that 'The great rule of inference' is that 'so far as there 



§§ 551-554] MISS JONES' VIEW. 327 

exists sameness, identity or likeness, what is trne of one thing 
will be true of another,' I do not think he helps us much. For 
in any purely formal affirmative inference by categorical syl- 
logism, it is not 'two things' that are named by the terms in 
each proposition, but one thing or group; and in a whole syl- 
logism not two things, or three things, or six things, but one 
thing, or one thing and part of that same thing." 

551. The reader will recall the statement made in the early 
part of the book, that in a proposition the subject and predi- 
cate were simply names for one thing. I am glad to know that 
so clear a thinker as Miss Jones agrees with me. 

552. She further says: "The denomination and the appli- 
cation of the two terms in any affirmative proposition must be 
absolutely identical, and where there are more than three 
terms in an affirmative syllogism, the extra ones must be iden- 
tical in denomination with part of the denomination of some 
of the three, e. g., in 

All N's are Q's 

Some R's are N's, Some R's are Q'3 
we have four terms, viz.: 

(1) All N's 

(2) Some Q's 

(3) Some N's 

(4) Some R's, 

but we have not three or more things, but one group of things, 
viz. the Q's that are N's and a group that is all or a part of 
this, the R's that are N's." 

553. And again, I am glad to call attention to the fact that 
Miss Jones' position agrees with mine when she says that "the 
two terms (i. e. the subject and predicate) in any affirmative 
proposition must be absolutely identical." The reader will 
remember that we have repeatedly called attention to the fact 
that in stating propositions we must so state them thai they 
will read backward or forward. 

554. But is Miss Jones consistent when she savs, "And ii» 



328 SYLLOGISM. [Chap. 20. 

negative propositions and syllogisms we have only two things 
or two things and a part of one of theni?" 

In our system it makes no difference in this respect, whether 
a proposition is affirmative or negative, the subject and predicate 
must be identical and must both be names for the same thing, 
otherwise there could not be such a thing as a proposition. In 
making a proposition the mind holds in its grasp a thing or a 
group of things, and in order to describe it, or express a judg- 
ment in regard to it, the mind gives it one name and then 
another name and connects the two names together by the 
word "is" in a simple categorical proposition. It makes no 
difference whether the thing or the group of things, or the 
condition or state, is given an affirmative or negative name, 
the subject and the predicate, that is, the names of the thing 
or the condition or the state or whatever it is that the mind 
is holding in its grasp, must necessarily refer to the same 
thing, condition, or state, etc. The mind cannot be occupied 
with two things at once, any more than physically the same 
thing can occupy different places at the same time, or, to put 
it in other words, two bodies cannot occupy the same space at 
the same time. 

555. Again, she says: "Jevons; rule has the further fault 
of being reducible to tautology, for, so far as there exists 
sameness, identity or likeness, what is true of one thing will be 
true also of another," can only mean "two things that are like, 
in as far as they are like" (two things cannot be identical or 
the same). 

In this rule some of the important terms are in themselves 
ambiguous, and they are very loosely used. In employing like- 
ness, identity, sameness, what is true, as he does in his canon, 
Jevons errs, it seems to me, in two ways: 

(1) He confuses things that differ (qualitive likeness and 
quantitive identity — this confusion runs through his whole 
account of Inference); 

(2) His phraseology implies a distinction where there is no 
difference, and thus a real tautology wears the guise of signifi- 
cant assertion. 



g 556.] MISS JONES ON JEVON'S THEORY. 329 

When (Op. CM., p. 10,) he says, "In speaking of measuring 
extended objects, that we obviously employ the axiom that 
whatever is true of a thing as regards its length, is true of its 
equal in length, the absolute uselessness of the axiom seems 
clear — it amounts to no more than this, what is true of a given 
length in one case, is true of that length in another case, but 
this is only equivalent to saying that a given length is a given 
length, a form of words which has no predicative force. The 
matter is not mended by the further discussion (Op. Cit.< p. 17, 
and following pages) of logical inference, which appears to me 
to be spoiled all through by the ever-recurring confusion, 

(1) Between quantitive identity (what Jevons would perhaps 
call numerical sameness or identity) and qualitive likeness; 

(2) Between what I have called Independent and Dependent 
Propositions (to which latter class all mathematical equations 
belong). 

There is perhaps also some confusion between terms and 
application (denotation) of terms. The implication that two 
things can be so similar point for point, as to be capable of 
being logically substituted one for another, seems due to (1)." 

Miss Jones here refers to the confusion between quantitive 
identity and qualitive likeness. I entirely agre^b with her that 
Prof. Jevons' principle of reasoning, which he calls the Substi- 
tution of Similars, is useless in logic. 

556. Again she says: "It seems to me that Jevons was con- 
stantly on the very verge of escaping from the first of the confu- 
sions indicated above, but that somehow he always just missed 
doing so." 



CHAPTER XXI. 



THE FIGURES. 



557. The figures of the syllogism are merely different ways 
of stating it; just as the same act of reasoning may be stated 
categorically or hypothetically, so also can the syllogism be 
expressed in either of the four figures. 

558. First Figure. 

No true lover of pleasure is a true philosopher, 
The Epicureans were lovers of pleasure, 
Therefore they were not true philosophers. 
Let A = true lover of pleasure, 
B = true philosopher, 
C = Epicureans. 
The premises can be stated thus: 

(1) NoA = B 

(2) C = CA. 

Now, if No A = B, then the combinations containing AB 
are inconsistent and we eliminate them by making a figure 1 
in those sections of an ABC Reasoning Frame; 



AB 


Ab 


aB 


ab 




1 




2 


2 


C 


1 








c 



Fiff. 235. 



Again, if C = CA, then the combinations containing Ca 
are inconsistent and we eliminate them by making a figure 2 
in those sections. 

From the combinations which remain we can get 



§§ 559, 560.] 



SECOND FIGURE. 



331 



C = Cb, which can be translated 
The Epicureans were not true philosophers. 

559. Second Figure. 

No true philosopher is a lover of pleasure, 
The Epicureans were lovers of pleasure, 
Therefore, they were not true philosophers. 
Let A = true philosopher, 
B = lovers of pleasure, 
C = Epicureans. 
The premises can be stated thus : 

(1) No A = E 

(2) C = CB 

Now, if No A = B, then the combinations containing AB 
are inconsistent and we eliminate them by making a figure 1 
in those sections of an ABC Seasoning Frame: 



AB 


Ab 


aB 


ab 




1 


2 




2 


C 


1 








c 



Fig. 230. 



Again, if C = CB, then the combinations containing Cb are 
inconsistent and we eliminate them by making a figure 2 in 
those sections. 

From the combinations which remain we can get, 
C = Ca which can be translated, 
The Epicureans were not true philosophers. 
560. Third Figure. 

No lover of pleasure is a true philosopher, 
Lovers of pleasure were Epicureans, 
Therefore the Epicureans were not true philosophers. 
Let A = lovers of pleasure, 
B = true philosophers, 



333 



FIGURES. 



[ Chap. 31. 



C = Epicureans. 
I assume that the proposition, Lovers of pleasure were Epi- 
cureans, means, 

Some lovers of pleasure were Epicureans. ' 
The premises can be stated thus: 

(1) No A = B 

(2) CA = 0. 

Make an ABC diagram: 



AB 


Ab 


aB 


ab 




1 




2 


2 


C 


1 








c 



Fig. 237. 

Now, if No A = B, then the combinations containing AB 
are inconsistent and we eliminate them by making a figure 1 
in those sections. 

Again, if 0= CA, then the combinations containing Ca are 
inconsistent and we eliminate them by making a figure 2 
in those sections. 

From the uneliminated combinations we can get this defini- 
tion of C, 

C == CAb, 
which can be translated, 

The Epicureans were lovers of pleasure and not true 

philosophers. 
The Epicureans were not true philosophers. 

561. Fourth Figure. 

No true philosopher is a lover of pleasure, 

Lovers of pleasure were Epicureans, 

Therefore the Epicureans were not true philosophers. 

Let A = true philosophers, 



§§ 562, 563.J 



FOURTH FIGURE. 



333 



B = lovers of pleasure. 
C = Epicureans, 
The premises can be stated thus : 

(1) No A = B 

(2) CB = C 

Make an ABC diagram : 



AB 


Ab 


aB 


ab 




1 


2 




2 


C 


1 








c 



Pig. 238. 

Now, if No A = B, then the combinations containing AB 
are inconsistent and we eliminate them by making a figure 1 
in those sections. 

Again, if C = B, then the combinations containing Cb 
are inconsistent and we eliminate them by making a figure 2 
in those sections. 

From the uneliminated combinations we can get this defini- 
tion of C, 

C = CaB, 
which can be translated, 

The Epicureans were lovers of pleasure and not true 
philosophers. 
5G2. Let P stand for the major term ; M for the middle term ; 
S for the minor term, then we can state the four figures thus: 
Fig. 1. Fig. 2. Fig. 3. Fig. 4. 

M = P P = M M = P P = M 

S = M B = M M . = S M = S 

563. Speaking of the special rules of the figures and the 
determination of the legitimate moods in each figure, Dr. 
Keynes, in "Formal Logic," p. 2G4, says: 



334 FIGURES. [Chap. 21. 

"It may first of all be shown that certain combinations of the 
premises are incapable of yielding a valid conclusion in any 
figure." 

564. From two particular propositions no conclusions can 
be drawn. Particular propositions which must be stated in 
the form of "AB = AB" have no inconsistents in the Reason- 
ing Frame, and when nothing can be eliminated from the 
Reasoning Frame no conclusions can be drawn. It seems to 
me that the word "some" should be banished from the vocab- 
ulary of those who wish to reason exactly. 

565. Let A = universal affirmative 

I = particular affirmative 
E = universal negative 
O = particular negative 
"Then there can be sixteen combinations of premises, the 
major premises being stated first, thus: 

AA IA EA OA 

AI II EI 01 

AE IE EE OE 

AO 10 EO 00 

But, according to the rules of the syllogism, EE, OE, 00, 
which contain two negative premises, yield no conclusions, and 
II, 10, 01, which contain two particular premises, yield no con- 
clusions." 

566. According to the special rules of the syllogism, in Fig. 
1, the minor premise must be affirmative, (2) The major premise 
must be universal. 

These rules leave four combinations in Fig. 1, viz: AA, AI, 
EA, EI. AA will yield the conclusion A or I; EA either E or 
•0; AI only I; EI only O. This leaves six moods, which do not 
offend against any of the rules of the syllogism, viz: AAA, 
AAI, All, EAE, EAO, EIO. The last letter in each combina- 
tion stands for the conclusion. 

567. Dr. Keynes further says, p. 270 : "In this figure (Fig. 
1) it is possible to prove conclusions of all the forms, A, E, I, O, 



508.] 



FIGURE 2. 



33, 



and it is the only figure in which a universal conclusion can be 
proved. 

In Fig. 2 only negatives can be proved, and, therefore, it is 
chiefly used for purposes of disproof, for example: 

Every real natural poem is naive. 

Those poems of Ossian which Macpherson pretended to dis- 
cover are not naive, but sentimental. Hence they are not real 
natural poems." 

Let A = real natural poems 
B = naive 
C = poems of Ossian, etc. 

The premises can be stated thus : ' 

(1) A = AB 

(2) C = Cb 

Now if A = AB, then the combinations containing Ab are 
inconsistent, and we eliminate them by making a figure 1 in 
those sections of an ABC Reasoning Frame: 



AB 


Ab 

1 

1 


aB 
2 


ab 









C 


c 



Fig. 239. 

Again, if C = Cb, then the combinations containing CB are 
inconsistent, and we eliminate them by making a figure 2 in 
those sections. From the combinations which remain we can 
get this definition: 
C = Ca 
which can be translated, 

Those poems of Ossian, etc., are not real natural poems. 

568. Figure 2 is also called the exclusive figure because it is 
used to successively exclude various suppositions. This pro- 
cess is called abscissio infiniti. 



336 



FIGURES. 



[Chap. 21. 



569. Let us take this example of the third figure: 

Socrates is wise 

Socrates is a philosopher 
Therefore, some philosophers are wise. 
Let A — Socrates 

B === wise 

C = philosopher. 
The proposition can be stated thus: 

(1) A = AB 

(2) A = AC. 

Now, if A = AB, then the combinations containing Ab are 
inconsistent, and we eliminate them by making a figure 1 in 
those sections of an ABC Reasoning Frame: 



AB 


Ab 


aB 


ab 






1 






C 


2 


12 






c 



Fig. 240. 

Again, if A = AC, then the combinations containing Ac are 
inconsistent, and we eliminate them by making a figure 2 in 
those sections. From the combinations which remain we can 
get the following definition of C: 

C=CAB | CaB | Cab 
which can be translated, 

Philosophers are either Socrates and wise, or not-Soc- 
rates and wise, or not-Socrates and not-wise. 
I suppose the old logic would translate it in this way: 

Some philosophers are wise and some philosophers are 

not wise; 
Some philosophers are Socrates and some philosophers 
are not Socrates. 
These conclusions are extra-logical. 



CHAPTER XXII. 

THE MOODS. 

570. There are three propositions in a syllogism and they 
differ in quantity and quality. 

These differences are indicated by the symbols A, E, I, O, 
which stand respectively for, 

Universal Affirmative, Universal Negative, 
Particular Affirmative, Particular Negative. 
These differences are said to determine the mood of the 
syllogism. Now, as there are four kinds of propositions, and 
three propositions in each syllogism, there can be sixty-four 
different combinations. 

571. If any one of the four of the above-named kinds of pro- 
positions A, E, I, O, be the major premise, each one of these 
majors may have four different minor premises, and these six- 
teen pairs of premises may each have four different conclu- 
sions, and four time four times four equals sixty-four. But 
the rules of the syllogism reject the moods which have negative 
premises, and particular premises, and some moods for other 
faults, so that out of the sixty-four possible moods the syllo- 
gism allows only eleven, viz.: 

AAA, AAI, AEE, AEO, All, AOO, EAI, EAO, EIO, IAI, 
OAO. 

572. The moods of the syllogism which are allowed as men- 
tioned in the preceding paragraph, are not allowable in every 
figure, because a mood might violate some of the rules of the 
syllogism in one figure and not in another. By applying the 
moods to each figure, the old logicians have found that each 
figure will admit six moods only, but several of these are use- 
less because they draw a particular conclusion when a univer- 
sal might have been drawn. 

573. Let us take this example: 

All human creatures are entitled to liberty, 

£2 



338 



MOODS. 



[ Chap. 22. 



All slaves are human creatures, 
Therefore some slaves are entitled to liberty. 
Let A = all human creatures, 
B == entitled to liberty, 
C = all slaves. 
The premises can be stated thus: 

(1) A = AB 

(2) C = CA 

Now, if A = AB, then the combinations containing Ab are 
inconsistent, and we eliminate them by making a figure 1 in 
those sections of an ABC Reasoning Frame: 



AB 


Ab 


aB 


ab 






1 


2 


2 


C 




1 






c 



Fig. 241. 

Again, if C = CA, then the combinations containing Ca are 
inconsistent, and we eliminate them by making a figure 2 in 
those sections. 

From the combinations which remain we can get the follow- 
ing definition of C: 

C = CAB, which can be translated, 
All slaves are human creatures and entitled to liberty. 

574. The conclusion drawn by the old logic is not always 
the strict logical conclusion which necessarily follows from 
the premises. Five moods out of the twenty-four are neglected 
because they have particular conclusions when universals 
might have been drawn by the old logic. 

575. For the remaining nineteen moods names have been 
devised to distinguish the mood and its figure. In these names 
the three vowels represent the quality and quantity of the 
propositions and the consonants represent the figures. 



§§ 576, 577.] 



BARBARA. 



339 



576. The first figure has four moods. The first mood is 
AAA, and its name is Barbara ; in a certain way this is an arbi- 
trary designation; it has no meaning except to point out a cer- 
tain mood. The three A's in it mean that the mood is com- 
posed of three universal affirmative propositions. 
An example is: 

All men are fallible, 
All kings are men. 
Therefore all kings are fallible. 
Let A = All men, 
B = fallible, 
C = kings. 
The premises can be stated thus : 

(1) A = AB 

(2) C *= CA 

Now, if A = AB, then the combinations containing Ab are 
inconsistent, and we eliminate them by making a figure 1 in 
those sections of an ABC Reasoning Frame: 



AB 


Ab 


aB 


ab 






1 


2 


2 


C 




1 






c 



Fig. 242. 

Again, if C = CA, then the combinations containing Ca are 
inconsistent, and we eliminate them by making a figure 2 in 
those sections. 

From the combinations which remain we can get the follow- 
ing definition of C: 

C = CAB, which can be translated. 
All kings are fallible. 
577. The second mood of the first figure is EAE, and its 
name is Celarent. An example is: 



340 



MOODS. 



[Chap. 22. 



No men are gods, 
All kings are men, 
Therefore, no kings are gods. 
Let A = men, 
B == gods, 
G = kings. 
The premises can be stated thus : 

(1) No A = B 

(2) C = CA 

Now, if No A = B, then the combinations containing AB are 
inconsistent, and we eliminate them by making a figure 1 in 
those sections of an ABC Reasoning Frame: 



AB 


Ab 


aB 


ab 




1 




2 


2 


C 


1 








c 



Fig. 243. 

Again, if C = CA, then the combinations containing Ca are 
inconsistent, and we eliminate them by making a figure 2 in 
those sections. 

From the combinations which remain we can get the follow- 
ing definition of C : 

C = CAb, which we can translate, 
Kings are not gods. 
By reading the eliminated combinations, ABC, aBC, we can 

ffet 

"No C == B" 

which can be translated, 
No kings are gods. 
578. The third mood is All, and its name is Darii. An 
example is: 

All men are fallible, 
Some beiugs are men, 



579.] 



FERIO. 



341 



Therefore some beings are fallible. 
Let A = men, 
B = fallible, 
C = beings. 
The premises can be stated thus: 

(1)A = AB 
(2) C A == A 
Now, if A = AB, then the combinations containing Ab are 
inconsistent, and we eliminate them by making a figure 1 in 
those sections of an ABC Reasoning Frame: 



AB 


Ab 


aB 


ab 






1 






C 


2 


21 






c 



Fig. 244. 

Again, if CA = A, then the combinations containing Ac are 
inconsistent, and we eliminate them by making a figure 2 in 
those sections. 

From the combinations which remain we can get the follow- 
ing definition of C: 

G = CAB | CaB | Cab 
which we can translate, 

Beings are either fallible men, or fallible not-men, or 
not fallible not-men. 
By reading the eliminated combinations we can get : 
No A = b, 
which can be translated , 

No men are infallible; no infallibles are men. 

579. The fourth mood of the first figure is EIO, and its 
name is Ferio. An example is: 
No men are gods, 
Some beings are men, 



342 



MOODS. 



[ Chap. 22. 



Therefore some beings are not gods. 
Let A = men, 

B = gods, 

C = beings. 
The premises can be stated thus: 

(1) No A= B 

(2) CA = A 

Now, if No A = B, then the combinations containing AB 
are inconsistent, and we eliminate them by making a figure 1 
in those sections of an ABC Reasoning Frame: 



AB 


Ab 


aB 


ab 




1 








C 


12 


2 






c 



Fig. 245. 

Again, if A = C, then the combinations containing Ac are 
inconsistent, and we eliminate them by making a figure 2 in 
those sections. 

From the combinations which remain we can get the follow- 
ing definition of C : 

O == CAb | CaB | Cab 
which can be translated, 

Beings are either men and not gods; or gods and not 
men; or neither. 

580. The second figure has four moods. The first mood of 
the second figure is EAE, and its name is Cesare. An exam- 
ple is: 

No gods are men, 
All kings are men, 

Therefore no kings are gods. . » 

Let A = gods, 
B = men, 



§581.] 



CAMESTRES. 



343 



C = kings. 
The premises can be stated thus: 

(1) No A = B 

(2) C = CB 

Now, if No A = B, then the combinations containing AB are 
inconsistent, and we eliminate them bj making a figure 1 in 
those sections of an ABC Reasoning Frame; 



AB 


Ab 


aB 


ab 




1 


2 




2 


C 


1 








c 



Fig. 246. 

Again, if C = CB, then the combinations containing Cb are 
inconsistent, and we eliminate them by making a figure 2 in 
those sections. 

From the combinations which remain we can get the follow- 
ing definition of C: 

C == CaB 
which we can translate, 

Kings are not gods. 
By reading the eliminated combinations we can get y 
No B = A 
which we can translate, 
No men are gods. 
We can also get, 

No C = b, 
which we can translate, 

No kings are not-men. 

581. The second mood of the second figure is AEE, and its 
name is Camestres. 
An example is: 

All kings are men, 



344 



MOODS. 



[ Chap. 22. 



No gods are men, 

Therefore no gods are kings. , 

Let A = kings, 

B = men, 

= gods. 
The premises can be stated thus: 

(1) A = AB 

(2) No C = B 

Now, if A = AB, then the combinations containing Ab are 
inconsistent, and we eliminate them by making a figure 1 in 
those sections of an ABC Reasoning Frame: 



AB 


Ab 


aB 


ab 




2 


1 


2 




C 




1 






c 



Fig. 247. 

Again, if No = B, then the combinations containing CB 
are inconsistent, and we eliminate them by making a figure 2 
in those sections. 

From the combinations which remain we can get the follow- 
ing definitions: 

G = Cab 
which we can translate, 

Gods are not kings, 

A = ABo 
which we can translate, 
Kings are not gods. 
From the eliminated combinations we can get: 
No A == b 
which we can translate, 

No kings are not-men ; not-men are no kings. 
No C = A 



§§ 582, 583.] 



FESTINO. 



345 



which we can translate, 

No gods are kings; no kings are gods. 
No C = B 
which we can translate, 

No gods are men; no men are gods. 
Camestres varies but slightly from Celarent, EAE. 

582. An example of Celarent is: 

No men are gods, 
All kings are men, 
Therefore no kings are gods, 
which we have worked out in paragraph 577. 

583. The third mood of the second figure is EIO and its 
name is Festino. An example is: 

No gods are men, 

Some beings are men, 

Therefore some beings are not gods. 
Let A = gods, 

B = men, 

G = beings. 
The premises can be stated thus: 

(1) No A = B 

(2) CB = B 

Now, if No A = B, then all the combinations containing AB 
are inconsistent, and we eliminate them by making a figure 1 
in those sections of an ABC Reasoning Frame: 



AB 


Ab 


aB 


ab 




1 








C 


12 




2 




c 



Fig. 248. 
Again, if B = CB, then the combinations containing Be are 



346 



MOODS. 



[ Chap. 22. 



inconsistent, and we eliminate them by making a figure 2 in 
those sections. 

From the combinations which remain we can get the follow- 
ing definition of C: 

= CAb | CaB | Cab, 
which we can translate: 

Beings are either gods and not men, or men and not gods, or 
neither. 

584. The fourth mood of the second figure is AOO, and its 
name is Baroco. 
An example is: 

All gods are men, 
Some beings are not men, 
Therefore some beings are not gods. 
Let A = gods 
B = men 
= beings. 
The premises can be stated thus: 

(1) A = AB 

(2) Ob = Cb 

Now, if A = AB, then all the combinations containing Ab 
are inconsistent and we eliminate them by making a figure 1 
in those sections of an ABO Reasoning Frame: 



AB 


Ab 


aB 


ab 






1 






C 




1 






c 



Fig. 249. 



From the combinations which remain we can get the fol- 
lowing definition of C: 

C == CAB | CaB | Cab, 
which we can translate: 



§585.] 



DARAPTI. 



847 



Beings are either gods and men; or men and not gods; 
or neither. 
The eliminated combination Ab, can be read: 
No A = b ; 
which we can translate : 

No gods are not men; no not-men are gods. 
This mood gave the old logicians a good deal of trouble. 
They tried to show its validity by a process called the reductio 
ad impossible. They showed that the conclusion cannot be 
supposed to be false without contradicting one of the premises, 
and the premises are supposed to be true. 

585. The third figure has six moods. The first mood of the 
third figure is AAI, and its name is Darapti. 
An example is: 

All men are fallible, 
All men are living beings, 
Therefore, some living beings are fallible. 
Let A = men, 
B = fallible, 
C = living beings. 
The premises can be stated thus: 
A = AB 
A = AC 
Now, if A = AB, then the combinations containing Ab, are 
inconsistent and we eliminate them by making a figure 1 in 
those sections of an ABC Reasoning Frame: 



AB 


Ab 


aB 


ab 






1 






C 


2 


12 






c 



Fig. 250. 
Again, if A =AO, then the combinations containing Ac, 



348 MOODS. [ Chap. 22. 

are inconsistent and we eliminate them by making a figure 2 
in those sections. 

From the combinations which remain we can get the fol- 
lowing definition of C: 

C = CAB | CaB | Cab, 
which can be translated, 

Living beings are either men and fallible or not men and 
fallible; or neither. 
From the eliminated combinations we can get, 
No A = b 
which can be translated, 

No men are infallible; no infallibles are men. 
No A = c, 
which can be translated, 

No men are not-living beings; no not-living beings are 
men. 

586. The second mood of the third figure is IAI, and its 
name is Disamis. 
An example is: 

Some men are kings, 
All men are fallible beings, 
Therefore some fallible beings are kings. 
Let A = men 
B = kings 
C = fallible beings 
The premises can be stated thus: 

(1) AB = B 

(2) A = AC. 

Now, if B = AB, then the combinations containing aB 
are inconsistent and we eliminate them by making a figure 1 
in those sections. 

Again, if A = AC, then the combinations containing Ac 
are inconsistent and we eliminate them by making a figure 2 
in those sections of an ABC Reasoning Frame: 



§ 587.] 



DATISI. 



349 



AB Ab 


aB 


ab 








1 




C 


2 


2 


1 




c 



Fig. 251. 



From the combinations which remain we can get the fol- 
lowing definition of C: 

C = CAB | CAb | Cab, 
which can be translated, 

Fallible beings are men or kings or neither. 
From the eliminated combinations we can get, 

No A = c, 
which can be translated, 

No men are infallible beings; no infallible beings are 
men. I assume that not-fallible means infallible. 
No B = c, 
which can be translated, 

No kings are infallible. 

587. The third mood of the third figure is All, and its name 
is Datisi. An example is: 
All men are fallible 
Some men are kings, 
Therefore some kings are fallible beings. 
Let A = men 
B = fallible 
C = kings. 
The premises can be stated thus: 

(1) A = AB 

(2) AC = 0. 

Now, if A = AB. then all the combinations containing Ab 
are inconsistent and we eliminate them by making a figure 1 
in those sections of an ABC Reasoning Frame: 



350 



MOODS. 



[Chap. 22. 



AB Ab 


aB 


ab 




1 
| 1 


2 


2 


C 




1 






c 



Fig. 252. 



Now, if = AC, then the combinations containing aC are 
inconsistent and we eliminate them by making a figure 2 in 
those sections. 

From the combinations which remain we can get this defi- 
nition of C: 

C = CAB, 
which we can translate, 

Kings are fallible men. 

588. The fourth mood of the third figure is EAO, and its 
name is Felapton. 
An example is: 

No men are gods, 

All men are living beings 

Therefore some living beings are not gods. 
Let A = men, 

B — gods, 

C = living beings. 
The premises can be stated thus: 

(1) No A = B 

(2) A = AC 

Now, if No A = B, then all the combinations containing AB 
are inconsistent and we eliminate them by making a figure 1 
in those sections of an ABC Reasoning Frame: 



589.] 



BOCARDO. 



351 



AB 


Ab 


aB 


ab 




1 








C 


12 


2 






c 



Fig. 253. 

Again, if A = AC, then the combinations containing Ac 
are inconsistent and we eliminate them by making a figure 2 
in those sections. 

From the combinations which remain we can get the follow- 
ing definition of C: 

C = CAb | CaB | Cab, 
which we can translate: 

Living beings are either men and not gods, or gods and not 
men, or neither. 
The eliminated combinations can be read: 
No A = B, 
which we can translate, 

No gods are men, * 

No A = c 
which we can translate, 

No men are not-living beings; no not-living beings are 
men. 

589. The fifth mood of the third figure is OAO, and its name 
is Bocardo. An example is: 

Some men are not kings, 
All men are fallible, 
Therefore some fallible beings are not kings. 
Let A = men, 
B = kings, 
C == fallible. 
The premises can be stated thus: 

(1) Ab = Ab 

(2) A = AG 



352 



MOODS. 



[ Chap. 22. 



Ab == Ab has no contradictories. 

If A = AC, then all the combinations containing Ac are in- 
consistent and we eliminate them by making a figure 2 in 
those sections of an ABO Reasoning Frame. 



AB 


Ab 


aB 


ab 












C 


1 


1 






c 



Fig. 254. 

From the combinations which remain we can get the follow- 
ing definition of C : 

C = CAB | CAb | CaB | Cab, 
which we can translate: 

Fallible beings are fallible beings. 
The full translation of the definition of C, i. e., 

Fallible beings are men and kings, or men and not kings, 
or not-men and kings or neither, 
does not tell us anything more than, 

Fallible beings are fallible beings. 
The eliminated combinations can be read, 
No A = c, which we can translate: 
No men are not-fallible beings, i. e., 
No men are infallible, and, no infallible beings are men. 

590. The sixth mood of the third figure is EIO, and its name 
is Ferison. 
An example is: 

No men are gods, 

Some men are living beings, 

Therefore some living beings are not gods. 
Let A = men, 

B — gods, 

C = liviDg beings. 



§ 591.] 



BRAMANTIP. 



353 



The premises can be stated thus: 

(1) NoA = B 

(2) AC = AC 

Now, if No A = B, then the combinations containing AB 
are inconsistent and we eliminate them by making a figure 1 
in those sections of an ABC Reasoning Frame. 



AB 


Ab 


aB 


ab 




1 








C 


1 








c 



Fig. 255. 

AG = AC has no contradictories in the Reasoning Frame. 
From the combinations which remain we can get the follow- 
ing definition of C : 

C = CAb | CaB ] Cab, 
which we can translate: 

Living beings are either men and not gods, or gods and 
not men, or neither. 
591. The fourth figure has five moods. The first mood of 
the fourth figure is AAI, and its name is BramaDtip. An 
example is: 

All kings are men, 
All men are fallible, 

Therefore some fallible beings are kings 
Let A = kings, 
B = men, 
C = fallible. 
The premises can be stated thus: 

(1) A = AB 

(2) B = BC 

Now, if A = AB, then the combinations containing Ab are 
inconsistent and we eliminate them by making a figure 1 in 
those sections of an ABC Reasoning Frame: 
23 



354 



MOODS. 



[ Chap. 22. 



AB 


Ab 


aB 


ab 






1 






C 


2 


1 


2 




c 



Fig. 256. 

Again, if B = BO, then the combinations containing Be are 
inconsistent and we eliminate them by making a figure 2 in 
those sections. 

From the combinations which remain we can get the follow- 
ing definition of C : 



Fallible beings are either kings and men, or men and 
not-kings, or neither. 
From the eliminated combinations we can get: 
No A = b, 
which we can translate: 

No kings are not-men; not-men are no kings. 
No A — c 
which we can translate, 

No kings are infallible; no infallible beings are kings. 

592, The second mood of the fourth figure is AEE, and its 
name is Oamenes. An example is-- 

All kings are men 

No men are gods 

Therefore, no gods are kings. 
Let A = Kings, 

B = men 

C = gods 
The premises can be stated thus: 

(1) A = AB 

(2) No B = C 



593.] 



DIMARIS. 



355 



Now, if A = AB, then all the combinations containing Ab 
are inconsistent, and we eliminate them by making a figure 1 in 
those sections of an ABC Reasoning Frame: 



AB 


Ab 


aB 


ab 




2 


1 


2 




C 




1 




c 



Fig. 257. 



Again, if No B = C, then all the combinations containing BG 
are inconsistent, and we eliminate them by making a figure 2 
in those sections From the combinations which remain we 
can get the following definitions of C and c: 

C = Cab 
which we can translate, 
Gods are not kings 

c = cAB | caB | cab 
which we can translate, 

Whatever things are not-gods are kings and men, or men 
and not-kings, or neither. 
From the eliminated combinations we can get: 
No A = b 
which we can translate, 

No kings are not-men; no not-men are kings; 
No C = A 
which we can translate, 

No kings are gods; no gods are kings. 

593. The third mood of the fourth figure is IAI, and its 
name is Dimaris. An example is : 
Some living beings are men 
All men are fallible 
Therefore, some fallible objects are living beings. 



356 



MOODS. 



[ Chap. 22. 



Let A = living beings 

B = men 

C = fallible 
The premises can be stated thus: 

(1) AB = B 

(2) B = BO 

Now, if B = AB, then the combinations containing aB are 
inconsistent, and we eliminate them by making a figure 1 in 
those sections. 

If B = BO, then all the combinations containing Be are 
inconsistent, and we eliminate them by making a figure 2 in 
those sections of an ABC Seasoning Frame: 



AB 


Ab 


aB 

1 

12 


ab 










C 


2 






c 



Fig. 258. 

From the combinations which remain we can get the follow- 
ing definition of C : 

O = CAB | CAb | Cab 
which we can translate, 

Fallible beings are human living beings, or living beings 
not-men, or neither. 
The eliminated combinations can be read: 
No c = B 
which we can translate, 

No infallible beings are men; no men are infallible. 

594. The fourth mood of the fourth figure is EAO, and its 
name is Fesapo. An example is : 
No gods are men 
All men are living beings 
Therefore, some living beings are not gods. 



§594] 



FESAPO. 



357 



Let A = gods 

B = men 

C = living beings 
The premises can be stated thus: 

(1) No A = B 

(2) B ==, BO 

Now, if No A — B, then all the combinations containing AB 
are inconsistent, and we eliminate them by making a figure 1 
in those sections of an ABC Reasoning Frame: 



AB 


Ab 


aB 


ab 




1 






C 


1 
2 




2 




c 



Fig. 259. 



Again, if B = BC, then all the combinations containing 
Be are inconsistent and we eliminate them by making a figure 
2 in those sections. 

From the combinations which remain we can get the follow- 
ing definition of G: 

C = OAb | CaB | Cab, 
which we can translate: 

Living beings are either gods and not men, or men and 
not gods, or neither. 
From the eliminated combinations we can get, 

No A = B, 
which we can translate: 

No gods are men; no men are gods. 
No B = c, 
which we can translate: 

No men are not-living beings; no not-living beings are 
men. 



358 



MOODS. 



[ Chap. 22. 



595. The fifth mood of the fourth figure is EIO, and its 
name is Fresison. 
An example is: 

No gods are men, 

Some men are living beings, 

Therefore some living beings are not gods. 
Let A = gods, 

B = men, 

C = living beings. 
The premises can be stated thus : 

(1) NoA = B 

(2) BC == BO 

Now, if No A = B, then all the combinations containing 
AB are inconsistent and we eliminate them by making a figure 
1 in those sections of an ABC Reasoning Frame: 
BC == BC has no contradictories. 



AB 


Ab 


aB 


ab 




1 








C 


1 








c 



Fig. 260. 



From the combinations which remain we can get the follow- 
ing definition of C: 

C = CAb | CaB | Cab, 
which we can translate: 

Living beings are either gods and not men, or men and 
not gods, or neither. 
The eliminated combinations can be read, 
No A = B, which we can translate: 
No gods are men ; no men are gods. 
I think the reader will be ready to agree with me that many 
of these moods are trifling variations of the moods of the first 



§§ 596-599.] SPECIAL RULES. 359 

figure. The examples given are taken from Prof. Bain's 
"Deductive and Inductive Logic." 

596. The special rules for the legitimate moods of the sec- 
ond figure, are: 

(1) One premise must be negative, 

(2) The major premise must be universal. 

The special rules for the legitimate moods of the third figure 
are 

(1) The minor premise must be affirmative, 

(2) The conclusion must be particular. 

The special rules for the legitimate moods of the fourth fig- 
ure are: 

(1) If the major premise is affirmative, the minor pre- 
mise must be universal. 

(2) If either premise is negative, the major premise must 
be universal. 

(3) If the minor premise is affirmative, the conclusion 
must be particular. 

597. According to the old logic, wherever a universal con- 
clusion could be drawn, a particular conclusion might also 
be inferred. 

The old logic holds that "some" may be logically inferred 
from "all." I do not think this is logical. Logically the mind 
infers from positive to negative, or from negative to positive. 
Now "some" is neither synonymous with "all," nor do "all" 
and "some" stand in the relation of positive and negative to 
each other. 

The opposite of "all" is "not all;" the opposite of "some" is 
"not some." 

59S. The moods of the syllogism which allow of a particu- 
lar conclusion where a universal conclusion could have been 
drawn, are called Subaltern moods, and the conclusions are 
called Weakened conclusions. 

The Subaltern moods are of no practical importance and are 
generally omitted. 

599. A syllogism which has two universal premises from 
which a particular conclusion is drawn, is called a Strength- 



360 MOODS. [ Chap. 22. 

ened syllogism, with the exception of AEO in the fourth figure. 
The premises are stronger than they need to be to draw the 
particular conclusion. 
Thus if we had, 
Ail A. = B 
All A = 

Therefore some C = B 
the conclusion could have been obtained from the premises, 
Ail A = B 
Some A = C 
or from the premises, 
Some A = B 
All A = C. 

600. There are some forms which are not formal syllogisms 
in the strictest sense, though their correctness is immediately 
evident to most persons. Miss Jones gives this example: 
A is greater than B 
B is greater than 
Therefore A is greater than 0. 
This reasoning depends upon the assumption that, 

If B is greater than 0, what is greater than B is greater 
than 0. 
Let A =*= A 
B = B 

O = greater than B 
D = greater than 
The premises can be stated thus: 

(1) A = AG 

(2) B = BD 

(3) G = CD 

Now, if A = AG, then the combinations containing Ac are 
inconsistent and we eliminate them by making a figure 1 in 
those sections of an ABCD Reasoning Frame: 



§jj 601-604.] 



MNEMONIC LINES. 



361 



AB 


Ab 


aB 


ab 












CD 


23 


3 


23 


3 


Cd 


1 


1 






cD 


1 
2 


1 


2 




cd 



Fig. 261. 



Again, if B = BD, then the combinations containing Bd 
are inconsistent and we eliminate them by making a figure 2 
in those sections. 

Again, if C = CD, then the combinations containing Cd are 
inconsistent and we eliminate them by making a figure 3 in 
those sections. 

From the combinations which remain we can get the follow- 
ing definition of A : 

A == AD, which we can translate, 
A is greater than C. 

601. In order to aid the student to remember the nineteen 
valid moods of the syllogism certain curious lines have been 
invented. 

They are, 

Barbara, Cslarent, D&rli, FeriOque, prioris; Cesare, Cames- 
tres, Festin5, Bar ok 5, secundse; Tertia, Darapti, Disarms, 
Datisi, Felapton, Bocardo, Ferieon, habet; Quarta insuper 
addit Bramantip, Camenes, Dimaris, Fesapo, Fresison. 

602. The first line indicates the four moods of the first fig- 
ure. The second line indicates the four moods of the second 
figure. Then come the six moods of the third figure, and 
lastly the five moods of the fourth figure. 

603. The letter s indicates that the proposition indicated by 
the vowel preceding the letter s is to be converted simply. 

604. The letter p indicates that the proposition is to be con- 
verted per accidens or by limitation. 



302 MOODS. [ Chap. 22. 

605. The letter m which is derived from the Latin word 
mutare, which means to change, indicates that the premises 
of the syllogism are to be transposed. 

606. The letter k indicates that the mood must be proved 
by the process called the reductio ad impossible. 

607. The capital letters B, 0, D, F, which are the first conso- 
nants in the names of the moods of the first figure, indicate 
that the other moods beginning with the same initial letters 
are reducible to the mood of the first figure with the same 
initial letter. Thus, Cesare, Camestres, Camenes, are reduci- 
ble to Celarent, Darapti, Disamis, Datisi and Dimaris to Darii. 
Felapton, etc, to Ferio, and so forth. The foregoing rules are 
taken from Prof. Jevons. 

608. Aristotle did not recognize the fourth figure. It is 
supposed to have been invented by Galen, and hence it is fre- 
quently called the Galenian figure. Its use has been fre- 
quently condemned by the old logicians. 

Father Clark in his Logic, p. 337, says: 

"Ought we to retain it? If we do, it should be as a sort of 
syllogistic Hylot, to show how low the syllogism can fall when 
it neglects the laws on which all true reasoning is founded, 
and to exhibit it in the most degraded form which it can 
assume without being positively vicious. Is it capable of 
reformation? Not of reformation but of extinction. Where 
the same premises in the first figure would prove a universal 
affirmative, this feeble caricature of it, is content with a par- 
ticular; where the first figure draws its conclusion naturally 
and in accordance with the forms into which human thought 
instinctively shapes itself, this perverted abortion forces the 
mind to an awkward and clumsy process which rightly de- 
serves to be called 'inordinate and violent'." 

609. Dr. Keynes says, "Thomson's ground of rejection is, 
that 'in the fourth figure the order of thought is wholly inven- 
ted, the subject of the conclusion had only been a predi- 
cate, whilst the predicate had been the leading subject in the 
premise. Against this the mind rebels and we can ascertain 



§§ 610-616.] ENTHYMEMES. 365 

that the conclusion is only the converse of the real one, by 
proposing to ourselves similar sets of premises to which we 
shall always find ourselves in the- first figure, with the second 
premise first'." (Laws of Thought, p. 176). 

610. When one of the premises of a syllogism is omitted and 
it is understood, without being expressed, the syllogism is 
called Enthymeme; so also when the conclusion is left to be 
understood. If the major premise is omitted it is called an 
Enthymeme of the first order. 

611. If the minor premise is omitted it is called an Enthy- 
meme of the second order. 

612. If the conclusion is omitted it is called an Enthymeme 
of the third order. 

613. Dr. Keynes gives the following example of the three 
orders : 

First, Balbus is avaricious and therefore he is unhappy. 

Second, All avaricious persons are unhappy and there- 
fore Balbus is unhappy. 

Third, All avaricious persons are unhappy and Balbus 
is avaricious. 

614. A chain of syllogisms is called a Polysyllogism. 

615. In any syllogism, the conclusion of which is the pre- 
mise of a preceding syllogism, the preceding syllogism is called 
a Prosyllogism, and the one which contains the conclusion is 
called an Episyllogism. The same syllogism may be both an 
Episyllogism and a Prosyllogism. When the same syllogism 
proceeds from Prosyllogism to Episyllogism it is called pro- ' 
gressive. When the process is reversed, that is from Episyl- 
logism to Prosyllogism, then the reasoning is called regressive. 

616. An Epicheirema is a Polysyllogism in which one or 
more Prosyllogisms are merely indicated, thus: 

All B = D because it = 
All A = B 
Therefore all A = D 
This would be called a single Epicheirema. 



364 



MOODS. 



[ Chap. 22. 



The following is a double Epicheirema: 

All A = B because it = C 

All D = A because all E = A 

Therefore all D = B 
The above argument can be stated thus : 

(1) A = AB 

(2) A = AC 

(3) D = DA 

(4) E = EA 

Now, if A = AB, then the combinations containing Ab arc 
inconsistent and we eliminate them by making a figure 1 in 
those sections of an ABCDE Reasoning Frame. 



AB 


Ab 
1 


aB 


ab 






3 

4 


34 


CDE 




1 


3 


3 


CDe 




1 


4 


4 


CdE 




1 






Cde 


2 


1 
2 


43 


34 


cDE 


2 


1 
2 


3 


3 


cDe 


2 


1 


4 


4 


cdE 


2 


1 

2 






cde 



Fig. 2G2. 



§§ 617-621.] SORITES. 365 

Again, if A = AC, then the combinations containing Ac 
are inconsistent and we eliminate them by making a figure 2 
in those sections. 

Again, if D = DA, then the combinations containing Da 
are inconsistent and we eliminate them bj making a figure 3 
in those sections. 

Again, if E = EA, then the combinations containing Ea are 
inconsistent and we eliminate them by making a figure 4 in 
those sections. 

From the combinations which remain we can get the follow- 
ing definition of D: 

D = DB. 

617. Where in a Polysyllogism the conclusions, except the 
final one, are omitted, and each two successive propositions 
contain a common term, the argument is called a Sorites. 

618. There are two kinds of Sorites, the Aristotelian and 
the Gloclenian. In the Aristotelian the first premise contains 
the subject of the conclusion. In the Gloclenian the first pre- 
mise contains the predicate of the conclusion. 

619. The following is an example of the Aristotelian: 
All A = B 

All B = C 
All C = D 
All D = E 
Therefore all A = E. 

620. The following is an example of the Gloclenian: 
All D = E 

All C = D 
All B — C 
All A = B 
Therefore all A = E. 

621. Dr. Keynes gives the following rules of the Sorites: 
"(1) Only one premise can be negative; and if one is negative 

it must be the last. 
"(2) Only one premise can be particular; and if one is partic- 
ular it must be the first. 



366 MOODS. [ Chap. 22. 

"(1) There cannot be mor^e than one negative premise, for if 
there were since a negative premise in any syl- 
logism, necessitates a negative conclusion we 

should in analyzing the Sorites, somewhere come upon 
a syllogism containing two negative premises. Again if 
one premise is negative, the final conclusion must be 
negative, hence P must be distributed in this conclusion, 
therefore, it must be distributed in this premise, i. e., 
the last premise, which must accordingly be negative. 
If any premise, then, is negative, this is the one. 

"(2) Since it has been shown that all premises, except the 
last, must be affirmative, it is clear that if any, except 
the first, were particular, we should somewhere commit 
the fallacy of undistributed middle." 

622. In the examples given, the syllogisms in the Sorites 
are in Figure 1. A question has been raised as to whether 
there could be Sorites in Figures 2 or 3. Dr. Keynes says 
that there can be, and in this he is correct. He gives the 
following examples, (the lettering is mine): 

"Some A is not B 
C is B 
DisC 
E is D 

Therefore some A is not E." 

"Some D is not E 
D is G 
C is B 
B is A 
Therefore some A is not E." 

623. Dr. Keynes further says that the first Sorites given 
are in Figure 2 and in the mood Baroco, and the syllogism in 
the last Sorites given are in Figure 3 and in the mood Bocardo. 

624. The special rules given for the Sorites apply only to 
Figure 1. 



§ 625.] 



SORITES. 



367 



625. Let us take the following example of a Sorites: 

DE = DE 

D = DC 

C = CB 

B = BA 

Therefore AE = AE. 

DE == DE has no contradictories. 
If D = DC, then the combinations containing Dc are incon- 
sistent and we eliminate them by making a figure 1 in those 
sections of an ABCDE Reasoning Frame. 



AB 


Ab 


aB 


ab 






2 


3 


o 


CIE 
CDe 




2 


3 


2 




2 


3 


3 


CdE 




2 


3 


2 


Cde 


1 


1 


13 


1 


cDE 


1 


1 


13 


1 


cDe 






3 




cdE 






3 




cde 



Fig. 263. 



Again, if C = CB, then the combinations containing Cb 
are inconsistent and we eliminate them by making a figure 2 
in those sections. 

Again, if B = BA, then the combinations containing Ba, 
are inconsistent and we eliminate them by making a figure 3 
in those sections. 

From the combinations which remain we can get the follow- 
ing definition of AE: 

AE-= AE 



368 MOODS. [ Chap. 22. 

626. Dr. Keynes gives several examples like the preceding 
one, in which a particular conclusion is drawn. But as a par- 
ticular amounts to a tautologous proposition, such as A = A, 
the Sorites yielding only particular conclusions are of no conse- 
quence. 

627. We gave an example of an argument which read, 
A is greater than B 

B is greater than 

Therefore A is greater than C. 
There are an indefinite number of other arguments which 
have a somewhat similar form and can be worked out on paral- 
lel lines, thus: 

A equals B 

B equals C 

Therefore A equals C 
It is assumed here that if B equals C, whatever is equal to 
B, will be equal to the equal of B. Similar arguments are: 

A is a contemporary of B 

A is the brother of B 

A is to the right of B 

A is in tune with B 
and so on. 

628. These kinds of arguments are not syllogisms, and yet 
they are valid arguments. Yet Archbishop Whately says: 

"Syllogism is the form to which all correct reasoning may 
be ultimately reduced." 

629. Prof. Bay makes the following claim: 

"The syllogism is the type of all valid reasoning; for no 
reasoning will be valid unless it can be thrown into the form of 
a syllogism." 

630. Spalding says, 

"The syllogism is the norm of all inferences whose antece- 
dent is more complex, and all such inferences may by those 
who think it worth while, be resolved into a series of syllo- 
gisms." (Lo^ic, p. 158.) 



§ 631 .] SYLLOGISTIC CLAIMS. r 69 

631. J. S. Mill says: 

"All reasoning by which from general propositions pre- 
viously admitted other propositions equally or less general, 
are inferred, may be exhibited in some of the above forms/' 
i. e., the syllogistic moods. (Logic, p. 191.) 



24 



CHAPTER XXIII. 

PROPOSITIONS. 

632. We have already said more or less on the subject of 
propositions, but in this chapter we purpose to treat the subject 
a little more fully. 

A proposition is an act of the judgment giving two names 
to the one object in thought. The fundamental principle of 
the old logic that a proposition is the expression of the relation 
between the whole and its parts, is incorrect. 

G33. Every proposition has two terms, the subject and the 
predicate, which are connected by the copula "is." "Is" joins 
the two names which constitute the subject and the predicate 
together; it affirms the existence of the names or of the 
thoughts which are represented by the names. 

634. The theory of the old logic that "is not" is a copula 
and denies the predicate of the subject, is incorrect; the "not" 
is a part of the predicate name. 

635. Propositions are said to be opposed to each other 
when they have the same subject and the same predicate and 
at the same time the subject and predicate differ in quality 
or quantity or both. 

636. A universal affirmative and a universal negative hav- 
ing the same subject and predicate are termed by the old logic, 
Contraries. % 

637. A universal affirmative and a particular affirmative, 
a universal negative and a particular negative are called Sub- 
alterns. 

638. A particular affirmative and a particular negative are 
called Sub-contraries. Contraries and Sub-contraries differ in 
quality. 

639. Subalterns differ in quantity. 



§§ 640-647.] CONTRADICTORIES. 371 

040. Contradictories differ both in quality and quantity. 
Contradictories cannot both be true, for it cannot be true that 
"All men are mortal," and that "No man is mortal." 

Sub-contraries cannot both be false; "Some horses are 
black" and "Some horses are not black" are sub-contraries; 
both propositions cannot be false. 

C41. Subalterns both may be true or false, thus: 
All men are liable to mistakes, and, 
Some men are liable to mistakes, 
are both true. 

No men are liable to mistakes, and, 
Some men are not liable to mistakes, 
are both false. 

642. In Contradictories, if one is true, the other is false. 
If "All men are mortal" is true, then "No men are mortal" is 
false. 

643. Propositions are said to be converted when their terms 
are transposed. As, 

"Some cowards are boasters," 
"Some boasters are cowards." 

644. A universal negative can be converted simply: 
"No vegetables are stones, 

"No stones are vegetables." 

645. Particular affirmatives can be converted simply: 
"Some men are tall, 

"Some tall things are men." 

646. A particular negative must be treated as a particular 
affirmative: 

"Some members of the university are not learned," 

"Some not-learned are members of the university." 

This example illustrates very clearly the statement already 

made that the word "not" belongs to the predicate and not to 

the copula. 

647. An universal affirmative must have the quantity of the 
predicate in conversion explicitly stated, thus: 

"All birds are animals," 
"Some animals are birds." 



372 PROPOSITIONS. [Chap. 23. 

In the sentence "All birds are animals" it is implied, but not 
expressed, that all birds are some animals. 

648. A proposition is not true when any one of the state- 
ments which it contains is false, as: 

"Csesar was put to death in the 610th year of Kome, by 
those whose lives he spared when conquered." 
Csesar was put to death in the 710th year of Rome. 
The foregoing definitions are taken from the old logic, and 
in general are true. 

649. According to the old logic, all universal propositions 
distribute the subject. No particular propositions distribute 
the subject. All negatives distribute the predicate. No affir- 
matives distribute the predicate. 

I think this is a mistake. Usually a negative predicate is 
indefinite and undistributed. "Some men are not tall" means 
some men are some not tall things. "No men are immortal" 
means, all men are some not immortals, i. e,. some mortals. 

Take this example: 

"All men are all rational animals." 

The old logic said that it was merely accidental that "all 
rational animals" was a distributed term, and that it was not 
implied in the form of the expression. 

650. Prof. Bain says, "Every proposition must be either 
true or false, and so on the other hand, nothing else can be, 
strictly speaking* either true or false. In colloquial language, 
however, true and false are often more loosely applied, as when 
men speak of the true cause of anything, meaning the real 
cause; the true heir, that is, the rightful heir; a false prophet, 
that is, a pretended prophet. A true or false argument, mean- 
ing a valid or apparent argument. A man true or false to his 
friend, meaning faithful or unfaithful." 

651. Hobbes says, in his account of Categorical Propositions, 
that "the predicate is the name of the same thing of which the 
subject is a name," and Prof. Venn says, "What the state- 
ment (Plovers are lap-wings, clematis vitalba is travellers' 
joy) really means is that a certain object has two certain names 
belonging to it." 



§5 652 -654.] "IS" IN RELATION TO TIME. 373 

I am glad to be able to quote Hobbes and Prof. Venn on nty 
side of this question. 

G52. The copula "is" has no relation to time; it expresses 
merely the fact that a certain thing has two names. If any 
other tense of the verb "to be" is used in a proposition, it is 
either understood as being equivalent in meaning to the pres- 
ent tense and the difference of tense is regarded merely as a 
matter of grammatical propriety, or else if the idea of time 
modifies the sense of the whole proposition, then this fact is 
one of the terms, and we can express it by "at that time" or 
similar words, as "This man was honest," i. e., was honest at 
that time. 

G53. Some logicians have thought that the logical effect of 
"is" was equational, that is, it had the meaning of "is equal 
to." But as in a proposition we are only speaking of one thing 
the equational theory is not correct. If we were talking of 
1 wo different things, there would be some basis for the idea of 
equality. 

G54. The old logic places considerable stress on the order 
of the premises in the syllogism, and there has been a great 
deal of discussion on the subject. Aristotle held that the 
major premise containing the predicate of the conclusion 
should stand first; but logically it makes not the slightest 
difference which comes first. Of course we must state one 
after the other, but the order of statement can make no differ- 
ence in logical results. It seems to me that Aristotle's plan 
is not the best one. I like the order. 

A is B 
B isC 

Therefore A is C 
much better than I do the inverted order, 
All B's are C's 
All A's are B's 
Therefore all A's are C's 



3 74 PROPOSITIONS. [ Chap. 23. 

655. Prof. Bain in "Logic," p. 159, says that a syllogism 
with two singular premises is not a genuine syllogism. Thus : 

Socrates fought at Delium, 
Socrates was the master of Plato, 
Therefore the master of Plato fought at Delium. 
He says that the proposition, "Socrates was the master of 
Plato and fought at Delium" compounded out of the two pre- 
mises, is nothing more than a grammatical abbreviation.'-' 

In no way, therefore, can a syllogism with two singular pre- 
mises be viewed as a genuine syllogistic or deductive infer- 
ence.' If this is true, then I think no syllogisms are gen- 
uine, because we can always combine into one statement, the 
information contained in the two premises of a syllogism. 

656. Speaking of negation Prof. Bain says, in "Logic," p. 
57, "The negative of a real property or thing is also real. If 
negation be simply the remainder when one thing is subtracted 
from a universe containing more than one, such negation is 
no less a positive reality than the so-called positive, in fact, 
positive and negative must always be ready to change places. 

657. Prof. Jevons, in "Principles of Science," p. 63, speak- 
ing of negative propositions, says, "It would be a mistake, 
however, to suppose that the real occurrence of negative terms 
in both premises of a syllogism, renders them incapable of yield- 
ing a conclusion The old rule informed us that from two neg- 
ative premises no conclusion could be drawn. But it is a fact 
that the rule in this bare form does not hold universally true, 
and I am not aware that any precise explanation has been 
given of the condition under which it is or is not imperative.'' 

658. Referring to the above I would say that the old rule 
holds good in the case of particular propositions where both 
terms are indefinite. I do not know of any other cases where 
it holds good, but in the case stated above the rule makes no 
difference, because you cannot draw any definite conclusion 
from two wholly indefinite premises, no matter whether they 
are affirmative or negative. 

659. Prof. Jevons further says, "Consider the following 
example: 



§} 660-663.1 NAMES FOR PROPOSITIONS. 375 

Whatever is not metallic is not capable of powerful magnetic 

influence. 
Carbon is not metallic. 

Therefore carbon is not capable of powerful magnetic in- 
fluence. 
Here we have two distinctly negative premises, and yet they 
yield a perfectly valid conclusion. The syllogistic rule is 
actually falsified in its bare and general statement." 

660. The word "not" is the word commonly used for a neg- 
ative term, but the words "except," "omitting," "excluding," 
"but not," "only if not," have the same logical effect. 

Take the sentence "Lawyers, not chancery solicitors," and 
we could substitute for the word "not" "except," or " omitting'' 
or "excluding," or "but not," or "only if not," and the logical 
effect would be the same. 

661. I suggest that two propositions which when worked out 
in the Reasoning Frame can both be read be called Consistents. 
That when one destroys a combination necessary to the other 
they be called Inconsistents. That when they eliminate 
a letter-term they be called Contradictories. That when one 
eliminates every combination which the other saved and saves 
every combination which the other eliminated, they be called 
Perfect Contradictories. That when each eliminates and saves 
the same combinations that the other did, they be called Equiv- 
alents. 

662. The terms A and a are opposites but not contradictor- 
ies. These terms can stand for propositions and then one is 
a necessary inference of the other. Propositions may be con- 
tradictories, but terms cannot be. 

663. Given the proposition, 

All A = all B 
its negative equivalent is 

All b = all a. 



376 



PROPOSITIONS. 



[ Chap. 23. 



To take a concrete example, 

If all salt is all chloride of sodium, its negative equiva- 
lent is, 
Whatever is not chloride of sodium is not salt. 
Let us take the proposition, 

All A = some B, 
which can be stated thus: 

(1) A = AB 

Make an AB diagram: 



A 


a 








B 


1 




b 



Fig. 264. 

Now, if A = AB, then the combination Ab is inconsistent, 
and we eliminate it by making a figure 1 in that section. 

We can now read in the Reasoning Frame: 
b = ba 
and this is the negative equivalent of the given proposition. 

We can also read the following consistent propositions: 

(1) B '•'== A | a 

(2) a = B | b 

664. Not every consistent proposition is an inference, at 
least I make a distinction between consistent propositions and 
inferences. I call a consistent proposition one which can be 
read anywhere in the Reasoning Frame. 

665. An inference is a consistent proposition which can be 
read in the uneliminated combinations, and which also elimi- 
nates some of the combinations, and no others, eliminated by 
the principal proposition or inferend, as Miss Jones calls it. I 
think this is an important discovery. 



§ 6C5.J 



INFERENCES. 



377 



Let us take the proposition: 

A = ABC 
Make an ABC diagram : 



AB 


Ab 


aB 


ab 






1 






C 


1 


1 






c 



Fig. 2G5. 

Now, if A = ABC, then the combinations containing Ab, 
Ac, are inconsistent, and we eliminate them by making a figure 
1 in those sections. 
The following propositions are inferences: 
c = ca 
b = ba 
B = a | AC 
C = a | AB 
In this case the negative equivalent of the given proposition 
is a compound proposition, viz: 
b = ba and c = ca 
AB = ABC is an inference. 
A consistent combination is: 
be =bca 
this is also an inference, because it would eliminate one of the 
eliminated combinations and no one of the uneliminated com- 
binations. 
A contradictory proposition is: 
A = Be 
because it would cause the elimination of the letter A. 

Two propositions are independent when both can be read in 
the Reasoning Frame and each eliminates an entirely different 
set of combinations. 



378 



PROPOSITIONS. 



[ Chap. 23. 



666. Let us take the proposition: 

(1) AB = ABO 
Make an ABC diagram: 



AB 


Ab 


aB 


ab 















1 








c 



Fig. 266. 

Now, if AB = ABC, then the combination ABc is inconsist- 
ent, and we eliminate it by making a figure 1 in that section. 
We can now read: 

(1) c = ca | cAb 

This is the negative equivalent of the given proposition. 

(2) Be = Bca 

This is a consistent proposition but not an inference, because 
if the combination Bca was destroyed, it would not effect the 
expression of the given proposition. 

(3) BC = BCa 

is an inconsistent proposition, because it destroys the visible 
expression of the given proposition, that is, it eliminates a com- 
bination which is necessary to the expression of the given 
proposition. 

(4) A = ABc 

This is a contradictory proposition because it eliminates the 
letter A. 



CHAPTER XXIV. 

QUANTIFICATION. 

667. The predicates of propositions generally have no such 
words as "all" or "some" affixed to them to denote the distribu- 
tion or the non-distribution of the predicate name. And yet, 
of course, the predicate must always be meant to represent 
either "all" or "some." Whether expressed or not, it must 
really be either distributed or undistributed. 

668. Sir William Hamilton is the writer on logic who has 
most strenuously insisted on the quantification of the predi- 
cate. By quantifying the predicate we expressly state whether 
the predicate term means "all" or "some." 

Hamilton's rule was, "We must render explicit in the state- 
ment whatever is implicit in the thought." In nearly all argu- 
ments there are omissions and ellipses which must be supplied. 
Thus, for instance, the universal affirmative proposition, 

All A = B 
would be quantified and become, 

All A is some B 
which we would state symbolically, 

A == AB 

669. In our system we always quantify the predicate, and I 
think Hamilton is clearly right in insisting that before we can 
have accurate reasoning the predicate must be quantified. By 
quantifying the predicate, the old logic obtains four new 
moods, symbolized by U, Y, n and w. 

I have substituted n and w for the Greek letters which Prof. 
Hamilton used. 

U stands for "All S is all P." 



380 QUANTIFICATION. [Chap 24. 

Y stands for "Some S is all P." (P meaning predicate and 
S meaning subject), 
n stands for "No S is some P." 
w stands for "Some S is not some P." 

670. Some writers think that there should be recognized by 
the old logic, six forms of propositions, thus: 

A All S is P 

Y Only S is P, i. e., All P is some S 

E No S is P 

I Some S is P 

n Not only S is P, i. e No S is some P, Not S alone 

is P 
O Some Sis not P 

671. Dr. Keynes says, "Formal Logic," p. 334: "By a rigid 
quantification of the predicate, however, the distinction 
between subject and predicate may be dispensed with; and 
such being the case, there is no ground left for distinction of 
figure, which depends upon the position of the middle term as 
subject or predicate." 

672. The old logic uses the word "some" in the sense of 
"some and it may be all." In our system we use it in the sense 
of "some only." 

673. Prof. Lotze in his work on Logic, expresses some very 
correct views on the quantification of the predicate. On p. 59, 
he says : "When we say 'Gold is yellow/ it is indisputable that 
in this judgment our idea of gold lies within the sphere of yel- 
low, and that accordingly the predicate is of wider extent than 
the subject; but it certainly is not this that we intended to 
express by the judgment." 

Logic, indeed, has already drawn attention to the fact that 
we are not quite right even in making this sentence; appealing 
from what we express to what we mean, it teaches that the 
subject also, from its side, limits the too extensive predicate; 
gold is not yellow simply, but golden yellow, the rose is rosy 
red, and this particular rose only this particular rosy red. The 



g 673.] LOTZE"S VIEW 381 

relation which exists between them is primarily no more than 
this, that whenever or wherever, under certain conditions, the 
one idea, gold, is found, there the other idea, yellow, is also 
found, but that the former is not always present when the lat- 
ter is. 

We say, "Some men are black," and suppose ourselves to be 
making a synthetical judgment, because blackness is not con- 
tained in the concept of man. But the true subject of this sen- 
tence is not the universal concept man (for it is not that which 
is black), but certain individual men; these individuals, how- 
ever, though they are expressed as merely an indefinite portion 
of the whole of humanity, are yet by no means understood to 
be such an indefinite portion; for it is not left to our choice 
what individuals we will take out of the whole mass of men; 
our selection, which makes them 'Some' men, does not make 
them black, if they are not so without it; we have, then, to 
choose these men, and we mean all along only those men who 
are black, in short, negroes; these are the true subjects of the 
judgment. That the predicate is not meant in its universality, 
that on the contrary, only the particular black is meant which 
is found on human bodies, is at once clear, and I shall follow 
out this remark later; here I will only observe that it is merely 
the want of inflection in the German expression which deceives 
us as to its proper sense; the Latin 'Xon nulli homines sunt nigri' 
shows at once by number and gender that 'homines' has to be 
supplied to 'nigri.' The full sense, then, of the judgment is, 
'some men,' by whom, however, we are to understand only 
black men, 'are black men;' as regards the matter, it is per- 
fectly identical, and as regards its form, it is only synthetical 
because one and the same subject is expressed from two differ- 
ent points of view, as 'black men' in the predicate, as a frag- 
ment of all men in the subject." 

Again, we say, 'the dog drinks.' But the universal dog does 
not drink ; only a single definite dog, or many, or all single do^s, 
are the subject of the sentence. In the predicate, too, we mean 
something different from what we express; we do not think of 



382 QUANTIFICATION. [ Chap. 24. 

the dog as an ever-running syphon; he does not drink simply, 
always, and unceasingly, but now and then, and this 'now and 
then' also, though expressed as an indefinite number of 
moments, is not so meant; the dog drinks only at definite 
moments, when he is thirsty, or, at any rate, inclined, when he 
finds something to drink, when nobody stops him ; in short, the 
dog which we mean in this judgment, is really only the drink- 
ing dog, and the same drinking dog is also the predicate." 

Again, 'Caesar crossed the Rubicon;' but not the Caesar 
who lay in the cradle or was asleep, or was undecided what to 
do, but the Caesar who came out of Gaul, who was awake, con- 
scious of the situation, and who had made up his mind; in a 
word, the Caesar whom the subject of this judgment means is 
that Caesar only whom the predicate characterizes, the Caesar 
who is crossing the Rubicon, and in no previous moment of his 
life was he the subject to whom this predicate could have been 
attached. It is obvious, moreover, to every capacity, that 
when he had crossed the river he could not go on crossing it, 
but was across, so that in no subsequent moment of his life 
either can he be the subject intended in this judgment. 

I will give two more examples which Kant has made famous. 
It is said that the judgment, "A straight line is the shortest 
way between two points," is synthetical, for neither in the con- 
cept "straight," nor in that of "line," is there any suggestion of 
longitudinal measure. But the actual geometrical judgment 
does not say of a straight line in general, that it is the shortest 
way, but only of that one which is included between these two 
points. 

, Now this fact, the fact that its extension is bounded by two 
points, (and it is only with this qualification that it forms the 
true subject of the sentence), is the ground, in this certainly the 
satisfactory ground, for assigning the predicate to it. It is 
easy to see that the concept of a straight line, ab, between the 
points a and b, is perfectly identical with the concept of the 
distance of the two points; for we cannot give any other idea of 
what we mean by "distance in space" than this, and that it is 



§§ 674. 675.] LOTZE'S VIEW. 383 

the length of the Straight line between a and b. There is not, 
therefore, a shorter and longer distance between a and b, but 
only the one distance, ab, which is always the same. On the 
other hand, we can speak of shorter and longer ways between 
a and b; the concept of "way" implies merely any sort of pro- 
gression which leads from a /to b; as this requires the getting 
over of the difference which separates b from a, there can be 
no way leading from a to b which leaves any part of this differ- 
ence not got over; accordingly that the shortest of all possible 
ways is the distance, i. e., the straight line between the given 
points, is a judgment which as regards its matter is perfectly 
identical, and merely regards the same object from different 
aspects. 

Nor, again, can the arithmetical judgment 7 plus 5 = 12, 
because 12 is not contained in either 7 or 5; the complete sub- 
ject does not consist in either of the quantities singly, but in 
the combination of them required by the sign of addition ; but 
in this combination, if the equation is correct, the predicate 
must be wholly contained; the equation would be false if some 
unknown quantity had to be added to 7 plus 5 in order to pro- 
duce 12. Here, too, then, we have a perfectly identical judg- 
ment as regards its matter, and it is only synthetical formally, 
because it exhibits the same number 12 first, as the sum of 
7 plus 5." (p. 85). 

G74. There remarks of Lotze tend very strongly to confirm 
our theory that the subject and the predicate are namts of the 
same identical thing, and also to confirm our other theory that 
when we state propositions in symbolical language we should 
state them in the form of identical propositions, that is, prop- 
ositions which are true when read either forward or backward. 

G75. The reader will have seen that in our system we have 
no difficulty in treating particular propositions. But in 
"Studies in Logic," p. 47, the writer makes the following criti- 
cism on Boole's system: "The plan of treating a set of uni- 
versal premises as a command to exclude certain combinations 



384 QUANTIFICATION. [ Chap. 24. 

of the terms which enter them, is due to Boole; no adequate 
extension of his method, so as to take in particular proposi- 
tions, is possible, without the use of some device which shall be 
equivalent to a particular copula." 

676. I think that a particular copula is an impossibility. 
I am not familiar enough with Boole's system to say whether 
this criticism is just or not, but from what I have read of it, I 
cannot see any reason why it should not take in particular 
propositions. 

677. On page 124, "Elements of Logic," Miss Jones says: 
"Quantification of the predicate in categorical propositions, 
seems to me to occupy an impregnable position in logic, a posi- 
tion, however, very different from that assigned to it by Sir 
William Hamilton, Dr. Thomson, Prof. Baynes, and others. 
My opinion is, that while the traditional form of A, E, I, O 
propositions is to be retained, quantification is an indispen- 
sable instrument of conversion, and therefore of reduction. 
The place of quantification in logic is very curious, its function 
being often as completely hidden from those processes of con- 
version which involve it as the subterranean train in one of the 
loop-tunnels of the Swiss Alps would be to an observer who 
only saw it rush into one opening and emerge again in a few 
minutes from another, just above or just below. My meaning 
will be best elucidated by taking an ordinary proposition and 
tracing the changes which it undergoes in conversion. 

Let the proposition be: 

(1) All human beings are rational. 
The ordinary converse of this is : 

(2) Some rational creatures are human beings, or 

(3) Some rational creatures are human. 

(3) is perhaps the more perfect converse, because (1) and (3) 
resemble each other in having an adjectival term for P, while 
(2) has a substantive term for P. (1) and (3) are adjectival 
propositions, (2) a coincidental proposition. Adjectival prop- 
ositions cannot be converted 



§§ 678-680.] MISS JONES' VIEW. 385 

Again, she says: "If I alter the position of S and P in (1) as 
it stands, and say 

Rational are all human beings 
it is clear that conversion in the logical sense has not taken 
place; for rational is still the predicate and 'all human beings' 
is still the subject. The proposition has been merely turned 
round." 

678. In our logic the subject and predicate being names for 
the same thing, the only difference is in their position. The 
subject can become the predicate and the predicate the subject. 

679. Again, she says: "But it may be transformed to the 
equivalent coincidental proposition, 

(4) All human beings are rational creatures, 

and with this we can deal. It is not, however, any more than 
the adjectival (1) simply convertible. If altered into, 

Rational creatures are all human beings, 
the proposition thus obtained, besides being awkward, is 
ambiguous — it is by no means clear which term is to be taken 
as subject, and the 'all' might even be understood to qualify (or 
quantify) 'rational creatures.' 

The first step towards real conversion is taken when we pass 
from the unquantificated coincidental (4) to the quantificated 
proposition, 

(5) All human beings are some rational creatures. 
From this we go on to the quantificated converse, 

(6) Some rational creatures are all human beings, 
and from (6) to the unquantificated converse of (5), 

(7) Some rational creatures are human beings, 
From (7) we pass to the equivalent adjectival proposition, 
(8) Some rational creatures are human. 

680. Contrast this process with our method. 
Let us take the same example: 

All human beings are rational. 
Let A = human beings, 
B = rational. 

25 



386 



QUANTIFICATION. 



[Chap. 3 J, 



The proposition can be stated thus: 

A == AB 

Now, if A = AB, then the combination Ab is inconsistent, 
and we eliminate it by making a figure 1 in that section of an 
AB Reasoning Frame: 



A 


a 




1 




B 




b 



Fig. 267. 

Now, with us conversion is merely reading the result of the 
statement of the proposition in the Reasoning Frame. We can 
get the following: 

B = A | a, which we can translate, 
Rational beings are either human beings or not human 
beings. 
The old logic would translate this: 

Some rational beings are all human beings. 

681. Again, Miss Jones says: "In converting an E propo- 
sition, we should, I think, proceed as follows: Let the pro- 
position to be converted be, 

(1) No R is Q 

(2) Any R is not Q (by mere equivalence). Quantifi- 
cating (2) we get: Any R is not any Q (3), (3) converts to: 
Any Q is not any R (4). 

By disquantificating (4) we reach: 
(5) Any Q is not R, 
and (5) No Q is R (by equivalence)." 

682. By our system an eliminated combination reads back- 
ward or forward by simply prefixing the word "No" to the 
combination ; in other words, "No" represents the effect of the 
eliminating process. We can also insert it between the letters. 



§ 683.] MISS JONES' VIEW. 387 

In the example given by Miss Jones we would at once read, 
"No Q is R," Q is no R. 

683. It seems to me that Miss Jones does not believe in the 
Aristotelian theory of inclusion in a class, that is, that the 
subject is included in the predicate. She says, "When, for 
example, I say, "The sky is blue," my meaning, and my whole 
meaning is, that the sky has that particular color. I am not 
thinking of the class blue, as regards extension at all, I am not 
caring, not necessarily, what blue things there are, or if there 
is any blue thing except the sky. I am thinking only of the 
sensation of blue, and am judging that the sky produces this 
sensation in my sensitive faculty; or (to express the meaning 
in technical language) that the quality answering to the sensa- 
tion of blue or the power of exciting the sensation of blue, is an 
attribute of the sky. When, again, I say, All oxen ruminate, 
I have nothing to do with the predicate considered in exten- 
sion — the comprehension of the predicate — the attribute or set 
of attributes signified by it — are all that I have in mind; and 
the relation of this attribute, or these attributes, to the subject, 
is the entire matter of the judgment. 

When we say Phillip is a man, or, A herring is a fish, do the 
words "man" and "fish" signify anything to us but the bundle 
of attributes connoted by them? Do the propositions mean 
anything except that Phillip has the human attributes and a 
herring the piscine ones? Assuredly not. Any notion of a 
multitude of other men among whom Phillip is ranked, or a 
variety of fishes besides herrings, is foreign to the proposition." 

Again, Miss Jones says: "We have seen that propositions 
on their way to conversion, have to undergo the process of 
quantification. But the reason why O is pronounced incon- 
vertible is not because there is not any more difficulty in quan- 
tificating it than in quantificating the other propositions, bat 
because when the quantificated converse of O has been reached, 
the quantification of its predicate cannot be droppped without 
an illegitimate alternation of signification. For the com- 
monly accepted signification of the disquantificated converse 



B88 QUANTIFICATION. [ Chap 24. 

of O involves a quantification different from that which has 
been dropped — the dropped P — indicator being some, the 
P-indicator understood as involved in the unquantificated 
proposition reached by dropping it being any. And, as 
at the same time, ordinary thought and speech will not admit 
the explicitly quantificated form, it is inevitable that a logic 
which deals with J;he forms of ordinary thought and speech, 
should regard O as inconvertible. To take an instance: 
The proposition, 

(1) Some blackbirds are not black birds 
becomes by quantification, 

(2) Some blackbirds are not any black birds. 
This converts to, 

(3) Any black birds are not some blackbirds. 
Dropping the quantification of (3) we get : 

(4) Any black birds are not blackbirds, 
and this would be understood to mean, 

(5) Any black birds are not any blackbirds, 
(No black birds are blackbirds)." (p. 130.) 

Now, in our system, the proposition, 

Some blackbirds are not black birds means, 
blackbirds which are not black birds are not black birds and 
we state it symbolically thus: 
Ab=Ab 
and read it thus, 

not black birds are blackbirds. 

684. The quantification of the predicate led Sir William 
Hamilton to recognize eight different forms of propositions 
instead of the usual four: 

All S is all P 

All S is some P 

Some S is all P 

Some S is some P 

No S is any P 

No S is someP 

Some S is not any P 

Some S is not some P 



u 


i. e. 


A = B. B = 


A 


i. e. 


A == AB 


Y 


i. e. 


BA = B 


I 


i. e. 


AB = AB 


E 


i. e. 


No A = B 


n 


i. e. 


No A = AB 





i. e. 


Ab == Ab 


w 


i. e. 


Ab = Ab 



§§ 685 6S8.J ADVANTAGES OF QUANTIFICATION. 389 

685. In our system we are not compelled to have propo- 
sitions given to us in any specified form, because our system is 
able to deal with every kind of proposition in every kind of 
form, excepting numerical propositions. 

686. Prof. Hamilton's position that "In thought the predi- 
cate is always quantified," is correct, and hence it follows that, 
"in logic the quantity of the predicate must be expressed on 
demand, in language." 

687. Dr. Bain says, "The quantity of the predicate is not 
expressed in common language, because common language is 
elliptical. Whatever is not really necessary to the clear com- 
prehension of what is contained in thought, is usually elided in 
expression. But we must distinguish between the ends which 
are sought by common language and logic respectively. 
Whilst the former seeks to exhibit with clearness the matter 
of thought, the latter seeks to exhibit with exactness the form 
of thought. Therefore, in logic, the predicate must always be 
quantified." 

6S8. Dr. Keynes says, "Formal Logic," p. 168, "'Predication 
is nothing more nor less than the expression of the relation of 
quantity in which a notion stands to an individual, or two 
notions to each other. If this relation were indeterminate 
— if we were uncertain whether it was a part, or whole, or 
none — there could be no predication.' 

Amongst the practical advantages said to result from quan- 
tifying the predicate are the reduction of all species of the con- 
version of propositions to one, namely, simple conversion; and 
the simplification of the laws of syllogism. As regards the 
doctrine of the quantification of the predicate, the distinction 
between subject and predicate resolves itself into a difference 
in order of statement alone. Each propositional form can, 
without any alteration in meaning, be read either forwards or 
backwards, and every proposition may, therefore, rightly be 
said to be simply convertible. It is further argued that the 
new propositional forms resulting from the quantification of 



390 QUANTIFICATION. [Chap. 24. 

the predicate are required in order to express relations that 
cannot otherwise be so simply expressed. Thus U alone serves 
to express the fact that two classes are co-extensive and even 
w is said to be needed in logical divisions, since if we divide 
(say) Europeans into Englishmen, Frenchmen, etc., this 
requires us to think that some Europeans are not some Euro- 
peans (e. g., Englishmen are not Frenchmen)." 



On p. 172, Formal Logic, Dr. Keynes says, "It is alto- 
gether doubtful whether writers who have adopted the eight- 
fold scheme have themselves recognized the pitfalls surround- 
ing the use of the word "some." Many passages might be 
quoted in which they distinctly adopt the meaning — some but 
not all. Thus Thomson ('Laws of Thought/ p. 150), makes U 
and A inconsistent." 

690. Dr. Keynes says, p. 335, "(2) IUn in Fig. 1, is invalid, 
if some is used in its ordinary logical sense. The premises 
are: 

Some M is some P, 

All S is all M. 

We may therefore obtain the legitimate conclusions by sub- 
stituting S for M in the major premise. This yields, 
Some S is some P." 
Let us substitute A for S, B for M and for P. 
The premises can be stated thus : 

(1) BC = BO 

(2) A = B 

(3) B ==-- A 

BO = BC has no contradictories in the Reasoning Frame. 

If A = B, then the combinations containing Ab are incon- 
sistent, and we eliminate them by making a figure 1 in those 
sections of an ABC Reasoning Frame: 



§ 691.] 



SOME. 



391 



AB 


Ab 


aB 


ab 






1 


2 




C 




1 


2 




c 



Fig. 268. 



Again, if B = A, then all the combinations containing Ba 
are inconsistent, and we eliminate them by making a figure 2 
in those sections. 

From the combinations which remain we can get the follow- 
ing definition of A: 

AC | Ac = AC | Ac, which can be translated, 
according to the old logic, 

Some A is some C. 
He says next: "If, however, "some" is here used in the 
sense of "some only," No S is some P follows from some S is 
some P, and the original syllogism is valid, although a negative 
conclusion is obtained from two affirmative premises." 

From the eliminated combinations in the example given we 
can read: 

No A = AbC | Abe 

A translation of this would be, according to the old logic 

No A is some C 

691. Again, Dr. Keynes says: 

"(3) AYI in Figure 1, some being used in its ordinary logical 
sense, is equivalent to AAI in Figure 3, in the ordinary syllo- 
gistic scheme, and it is therefore valid, but it is invalid if some 
is used in the sense of some only, for the conclusion now implies 
that S and P are partially excluded from each other as well as 
partially coincident, whereas, this is not implied by the pre- 
mises. 



39a 



QUANTIFICATION. 



[ Chap. 24. 



692. AYI consists of three propositions in the form of, 
All A is some B, 
Some A is all B 
Therefore some A is some B 
Some A is all B can be read, 

All B is some A 
and the premises can be stated thus: 

(1) A = AB 

(2) B = BA 

Now, if A = AB, then the combination Ab is inconsistent, 
and we eliminate it by making a figure 1 in that section of an 
AB Reasoning Frame: 



AB 


Ab 


aB 


ab 






1 


2 




C 




1 


2 




c 



Fig. 269. 

Again, if B = BA, then the combination Ba is inconsistent, 
and we eliminate it by making a figure 2 in that section. 

From the combinations which remain we can get the follow- 
ing definition of A: 

A = B, i. e., All A = all B 

The conclusion "I" does not logically follow from the pre- 
mises A and Y. 

693. Dr. Keynes says, p. 177, of the proposition n, "This 
proposition in the form of No S is some P, is not, I think, ever 
found in ordinary use. We may, however, recognize its possi- 
bility; and it must be pointed out that a form of proposition 
which we do meet with, namely, Not only S is P, or Not S 
alone is P, is practically n, provided that we do not regard this 
proposition as implying that any S is certainly P." 



§§ 694, 695.] 



n PROPOSITIONS. 



393 



Let us work the example given, 
Let A = S 
B = P 
then the proposition can be stated : 

No A = AB 
Now, if No A =AB, then the combination AB is inconsist- 
ent, and we eliminate it by making a figure 1 in that section 
of an AB Reasoning Frame: 



A 

1 


a 




B 






b 



Fig. 270. 



The combinations which remain are, 
Ab, i. e., All A = some b, 
aB " Some a = all B, a only is B, 
ab " Some a = some b. 

694. Dr. Keynes further says that Archbishop Thomson 
remarks that n 'has the resemblance only and not the power 
of a denial. True though it is, it does not prevent our making 
another judgment of the affirmative kind, from the same 
terms.' " 

("Laws of Thought," p. 79). 

695. When we allow n to be represented by symbols, as in 
the preceding example, it denies the existence of one combina- 
tion, that is, eliminates it. He is right, however, when he says 
that it does not prevent our making another judgment of the 
affirmative kind, — that is, on our theory that all propositions 
are affirmative. 

The proposition that we obtained was, 

A = Ab 



394 



QUANTIFICATION. 



[ Chap. 24. 



Archbishop Thomson would probably call this "another 
judgment of the negative kind," because the old logic calls any 
proposition "negative" which contains a negative term. 

696. Dr. Keynes then says, "This is erroneous, for although 
A and n may be true together, U and n cannot, and Y and n 
are strictly contradictories." 

Let us take examples of these propositions: 

Let "A = AB" represent the proposition A, 

"No A = AB" represent the proposition n. 

Now, if A = AB, then the combination Ab is inconsistent, 
and we eliminate it by making a figure 1 in that section of an 
Ab diagram: 



A 


a 




2 




B 


1 




b 



Fig. 271. 

Again, if No A = AB, then the combination AB is inconsist- 
ent, and we eliminate it by making a figure 2 in that section. 

An examination of the Reasoning Frame now shows that all 
the A's are eliminated. This proves that the propositions A 
and n are contradictories, and that it is not possible for them 
to be true together. 

697. Let us take the propositions Y and n : 
Let AB = B represent the proposition Y, 

No A = AB represent the proposition n. 



§ 698.] U AND n PROPOSITIONS. 

Make an AB diagram: 



395 



A 


a 




2 


1 


B 






b 



Fig. 272. 

Now, if B = AB, then the combination Ba is inconsistent, 
and we eliminate it by making a figure 1 in that section. 

Again, if No A = AB, then the combination AB is inconsis- 
tent, and we eliminate it by making a figure 2 in that section. 

The result proves that Y and n are contradictories because 
the letter B is eliminated. 

698. Let us take the propositions U and n: 
Let A =B and B = A represent U, 
No A = AB represent n. 

Make an AB diagram: 



A 


a 




3 


2 


B 


1 




b 



Fig. 273. 

Now. if A = B, then the combination Ab is inconsistent, and 
we eliminate it by making a figure 1 in that section. 

Again, if B = A, then the combination Ba is inconsistent, 
and we eliminate it by making a figure 2 in that section. 

Again, if No A = AB, then the combination AB is inconsis- 
tent, and we eliminate it by making a figure 3 in that section. 



396 QUANTIFICATION. [ Chap. 24. 

The result proves that U and n are contradictories because 
the letter terms A and B are eliminated. 

699. Dr. Keynes says, (p. 178), 

"The proposition w is absolutely of no importance." And 
I think the same can usually be said of all particular affirm- 
ative propositions. 

700. The exclusive propositions, 

Only S is P, 
S alone is P, 
are examples of the propositional form called Y in the eight- 
fold scheme. They usually mean, 

Some S is all P 
If we let A stand for S and B for P, then by repeating the 
predicate in the subject, thus: 

AB = B, we can state the proposition Y so that it 
will be true when read backward. 
In working it we can say, If B = AB, then the combinations 
which contain aB are inconsistent, and we eliminate them, 
etc. 

701. According to our system, the eight propositional forms 
can be stated thus: 

IT All A is all B, A =B, B = A 

A All A is some B, A = AB 

Y Some A is all B, AB = B 

• I Some A is some B, AB = AB 

E No A is any B, No A = B 

n No A is some B, No A = AB 

O Some A is not any B Ab == Ab 

w Some A is not some B, Ab = Ab 

The proper stating of propositions is a very important mat- 
ter in logic. We should always bear in mind Hamilton's rule, 
"Render explicit in the statement, whatever is implicit in the 
thought." 



CHAPTER XXV. 

INCONSISTENCY. 

702. In oar system the total elimination of any letter is 
a certain sign of contradictoriness in the premises. 

703. According to the old logic, if a universal affirmative 
proposition is given, the universal negative is false, the par- 
ticular affirmative is true and the particular negative is false. 

If A is AB is true, 
Then, Xo A is B is false, 
and AB is AB is true, 
and Ab is Ab is false. 

704. If a universal negative proposition is given as true, 
then the universal affirmative proposition is false and the par- 
ticular affirmative proposition is false and the particular nega- 
tive proposition is true. 

If Xo A is B, 
then A is AB is false, 
aud AB is AB is false, 
and Ab is Ab is true. 

705. If the particular affirmative proposition is given as 
true, then, the universal affirmative proposition is unknown, 
the universal negative proposition is false and the particular 
negative proposition is unknown. 

If AB is AB, 

then, A is AB is unknown, 

and Xo A is B is false, 

and Ab is Ab is unknown. (?) 

700. If the particular negative proposition is given as true, 
then the universal affirmative proposition is false, the univer- 
sal negative proposition is unknown and the particular affir- 
mative proposition is unknown. 



398 INCONSISTENCY. [ Chap. 25. 

If Ab is Ab, 

then, A is AB is false, (?) 
and No A is B is unknown, 
and AB is AB is unknown. (?) 

707. If A is AB is false, 

then, No A is B is unknown, (?) 
and AB is AB is unknown, (?) 
and Ab is Ab is true. 

708. If No A is B is false, 

then, A is AB is unknown, (?) 

and AB is AB is true, 

and Ab is Ab is unknown. (?) 

709. If AB is AB is false, 
then, A is AB is false, 
and No A is B is true, 
and Ab is Ab is true. 

710. If Ab is Ab is false, 
then A is AB is true, 
and No A is B is false, 
and AB is AB is true. 

I doubt the correctness of the conclusions marked with a (?). 

711. The denial of a truth of a proposition is equivalent to 
the affirmation of the truth of its contradictory, and vice versa. 
Of two contradictory propositions, one must be true and the 
orher false. 

732. In our system, given any proposition as true, we elimi- 
nate the inconsistent propositions. In order to do this there 
must be no ambiguity in the proposition which is given to us. 
If we do not fully understand the meaning of the given propo- 
sition, we cannot tell what it denies. The meaning of a pro- 
position depends upon what it denies. 

713. A compound proposition or a complex proposition, 
usually has a large number of inconsistent propositions. If 
any one of the inconsistents is true, then, the given proposition 
is not true. 



§g 714-720.] EQUIVALENTS. 399 

714. If the contradictories of a given proposition contain a 
common term, they may be converted into a single alternative 
proposition, and then the single alternative proposition would 
be the contradictory of the given proposition. 

715. When we have two propositions like those which fol- 
low composed of the same terms or of their opposites, the two 
propositions are equivalent, and, consequently, if one is true, 
the other is also true. Thus, if the proposition, 

All A = All B 
is true, then the proposition, 

All a = all b 
is also true. 

716. If the proposition, 

A = AB 
is true, then the proposition, 

b = ba 
is also true. 

717. If the proposition, 

a = ab 
is true, then the proposition, 

B = BA 
is also true. 

718. Equivalent propositions are sometimes called equipol- 
lent, meaning equal in signification and logical force. 

719. The old logic says that the particular affirmative prop- 
osition, 

AB-AB 
is inferrible from the universal affirmative proposition, 

"All A is B" 
I think this is extra logical. 

720. A proposition is inconsistent with a given proposition 
w r hen it eliminates any combination which is necessary to the 
expression of the given proposition in the Reasoning Frame. 

Let the given proposition be, 

C = A I B 



400 INCONSISTENCY. 

Make an ABC diagram: 



[Chap. 25. 



AB 


Ab 


aB 


ab 




1 






1 


C 










c 



Fig. 274. 

Now, if C = A | B, then the combinations CAB, Cab are 
inconsistent, and we eliminate them by making a figure 1 in 
those sections. 

An examination of the Reasoning Frame now shows that 
either of the propositions, 

(1) Ab = Abe 

(2) aB = aBc 

(3) A = Ac 

(4) a = ac 

(5) B = Be 

(6) b = be 

is inconsistent with the given proposition, because it would 
eliminate one or the other of the combinations AbC, aBC, and 
these combinations are necessary to the expression of the given 
proposition. 

721. By the use of the Reasoning Frame I have discovered 
an easy method of finding propositions inconsistent with the 
given proposition. The method is as follows: 

First, make the visible expression of the given proposition in 
the Reasoning Frame, by eliminating the inconsistent combina- 
tions. 

Second, make a similar Reasoning Frame and eliminate in 
it the combinations which the given propositions saved, and 
which are necessary to its expression. 



§ 722.] 



AN EXAMPLE. 



401 



Third, from the uneliminated combinations in the Second 
Reasoning Frame, get definitions of the letter-terms and each 
of these definitions will be inconsistent with the given proposi- 
tion, because it will eliminate a combination which is neces- 
sary to the expression of the given proposition. 

722. Let the given proposition be, 

C = A | B 

Make an ABC diagram: 



AB 


Ab 


aB 


ab 




1 






1 


C 










c 



Fig. 275. 

Now, if C = A or B, then the combinations ABC, abC, are 
inconsistent, and we eliminate them by making a figure 1 in 
those sections. 

Make another ABC diagram: 



AB 


Ab 


aB 


ab 






1 


1 




C 










c 



Fig. 276. 

Eliminate the combinations AbC, aBC, by making a figure 1 
in those sections. 

From the uneliminated combinations we can get these defini- 
tions of the different letter-terms: 
26 



402 



INCONSISTENCY. 



[ Chap. 25. 



(1) A = AB | Abe 

(2) B = BA | Bac 

(3) a = ab | aBc 

(4) b = ba J bAc 

(5) = CAB | Cab. 

Each one of these definitions is inconsistent with the given 
proposition, because it will destroy a combination necessary 
to the expression of the given proposition, for example, 

A = AB | Abe. 
Make an ABC diagram : 



AB 


Ab 


aB 


ab 






1 






C 










c 



Fig. 277. 

Now, if A = AB | Abe, then the combination containing 
AbC is inconsistent and we eliminate it by making a figure 1 
in that section. 

The Reasoning Frame now shows that the combination AbO 
is eliminated. This was a necessary combination to the expres- 
sion of the given proposition, hence, the definition A = AB | 
Abe, is inconsistent with the given proposition. 

723. Let us take this example : 

The powers not delegated to the United States by this Consti- 
tution nor prohibited by it to the states, are reserved to the 
states respectively, or to the people. 

Let A = the powers delegated to the U. S., 
B = the powers prohibited to the states, 
C = the powers reserved to the states, 
D = the powers reserved to the people. 



§ 723.] 



THE TENTH AMENDMENT!. 



m 



The proposition can be stated thus; 

(1) ab = Cd | cD 

(2) Cd | cD = ab 

(3) C = Cd. 

Make an A BCD diagram: 



AB 


Ab 


aB 


ab 




2 


2 


2 


21 


CD 


1 


1 


1 




Cd 


1 


1 


1 




cD 








1 


cd 



Fig. 278. 

Now, if ab = Cd | cD, and if Cd | cD = ab, then the com- 
binations containing ACd, aBCd, abCD, AcD, aBcD, abed, are 
inconsistent and we eliminate them by making a figure 1 in 
those sections. 

Again, if C = Cd, then the combinations containing CD are 
inconsistent and we eliminate them by making a figure 2 in 
those sections. 

Make another ABCD diagram: 



AB 


Ab 


aB 


ab 












CD 








1 


Cd 








1 


cD 


1 


1 


1 




cd 



Fig. 279. 



404 INCONSISTENCY. ^ [ Chap. 25. 

Eliminate the combinations containing Acd, aBcd, abCd, 
abcD, by making a figure 1 in those sections. 

In this diagram we have eliminated the combinations which 
the given propositions saved. 

From the uneliminated combinations we can get the fol- 
lowing definitions of the different letter-terms: 

(1) A = AC | AcD, which can be translated: 

The powers delegated to the United States are either 
reserved to the states or reserved to the people. 

(2) B = BO ! BcD, which can be translated: 

The powers prohibited to the states are either reserved 
to the states, or to the people. 

(3) C = CD | CA | CaB, which can be translated: 

The powers reserved to the states are either reserved 
to the people, or delegated to the U. S., or prohibited 
to the states. 

(4) D = DC | DA | DaB, which can be translated: 

The powers reserved to the people are either reserved 
to the states, or delegated to the United States, or 
prohibited to the states. 

(5) a = aBC | aBcD | abCD | abed, which can be trans- 
lated : 

The powers not delegated to the United States are either 
prohibited to the states and reserved to the states, or 
prohibited to the states and reserved to the people, 
or reserved to the states and to the people 
or are neither prohibited to the states nor reserved 
to the states nor reserved to the people. 

(6) b = bAC | bAcD | baCD | bacd, which can be trans- 
lated: 

The powers not prohibited to the states are either dele- 
gated to the United States and reserved to the states, 
or delegated to the United States, and reserved to the 
people or reserved to the states and to the people, or 
are neither delegated to the United States nor reserved 
to the states nor reserved to the people. 



§ 723.] 



THE TENTH AMENDMENT. 



405 



(7) c = cAD | caBD | cabd, which can be translated: 

The powers not reserved to the states are either dele- 
gated to the United States and reserved to the people, 
or prohibited to the states and reserved to the people, 
or are neither delegated to the United States nor pro- 
hibited to the states nor reserved to the people. 

(8) d = dAO | daBC | dabc, which can be translated: 

The powers not reserved to the people are either dele- 
gated to the United States and reserved to the states, 
or are prohibited to the states and reserved to the 
states, or are neither delegated to the United States 
nor prohibited to the states nor reserved to the states. 

Each and every one of these definitions is inconsistent with 
the given proposition, because it destroys a combination which 
is necessary to the expression of the given proposition. 

Let us take the definition of b, b = bAC I bAcD j 
baOD | bacd: 

Make an ABCD diagram: 



AB 


Ab 


aB 


ab 


CD 
Cd 
















1 








1 


cD 








cd 



Fig. 280. 

Now, if b = bAC | bAcD | baCD | bacd, then the combi- 
nations containing abCd, abcD, are inconsistent and we elim- 
inate them by making a figure 1 in those sections. 

The Reasoning Frame now shows that we have eliminated 
two combinations which were necessary to the expression of 
the given propositions. Hence, the definition of b is incon- 
sistent with the given proposition. 



406 INCONSISTENCY. [ Chap. 25. 

724. Another method of obtaining inconsistent propositions 
Is by prefixing the particle no to the subject, or predicate of an 
uneliminated combination, e. g., let the uneliminated combi 
nation be AB, i. e., A = B, then No A = B, No B = A, are 
inconsistent with A = B. 



CHAPTER XXVI. 

CONVERSION. 

725. When the terms of a proposition are transposed, i. e., 
when the subject is made the predicate and the predicate the 
subject, the proposition is said to be converted. The old logic 
had a great many rules for the conversion of propositions. 

In our system we state all propositions so that they will read 
backward as well as forward. Prof. Jevons calls propositions 
in this form, Identical propositions. An Identical proposition 
can be read either way without any rules. 

If all A is all B, all B must be all A. 

If twice two is four, then four is twice two. 

726. In our system we read our combinations in any order 
we please, without stopping to think whether there are any 
rules for converting subjects into predicates and predicates 
into subjects. We do not recognize any real distinction 
between subject and predicate, except in the matter of posi- 
tion, one must come before the other, that is all there is to it. 
Neither do we recognize any difference between so-called posi- 
tive and negative propositions. These are not logical but con- 
versational distinctions. 

727. In conversion, the original proposition is called the 
convertend, and the inferred proposition is called the converse. 

728. Dr. Keynes gives the two following rules: 

(1) The converse must be the same in quality as the con- 
vertend. 

(2) No term must be distributed in the converse unless it 
was distributed in the convertend. These rules apply to what 
is called simple conversion. Thus, I, Some A is some B, can be 
converted simply into, Some B is some A. Stated symbolic- 
ally, AB is AB can be read BA is BA. 



408 CONVERSION. [ Chap. 26. 

729. The proposition E, no A is B, can also be converted 
simply into No B is A. In our system the logical effect of No 
A is B, is the elimination of the AB combination. 

We translate the eliminating mark by "No" and then we can 
read the combination indifferently, 

No A is B, No B is A, A is no B, B is no A. 

In a similar way we can write the letter s in the section co»- 
taining the combination AB, to indicate that the combination 
represents the proposition, Some A is some B, and that we can 
read it either way, Some A is some B and some B is some A. 
This way of indicating the particular affirmative proposition is 
logically equivalent to the form we commonly use, viz. : 

AB is AB. 

730. In the case of the universal affirmative proposition A, 

All A is AB, 
we cannot infer 

All B is A, 
because, in the first proposition the B means some B, and is 
therefore undistributed, while in the second proposition it is 
distributed. So that in this case the converse would be, 

Some B is all A. 
This is called conversion per accidens or conversion by limi- 
tation. 

731. In the old logic the particular negative proposition, 
Some A is b, is said to be incapable of ordinary conversion, 
because some A is undistributed in the convertend, and if A 
became the predicate, it must be distributed, for it is a rule of 
theold logic, thatthe predicateof anegative proposition must be 
distributed. 

732. Aristotle proved the conversion of the universal nega- 
tive proposition, No A is B, as follows : No A is B, therefore, 
No B is A; for, if not, some individual B, say C is A; and hence, 
C is both A and B; but this is inconsistent with the given propo- 
sition, No A is B. 



§§ 733-730.] CONTRAPOSITION. 409 

733. Dr. Keynes proves the conversion per accidens, of the 
universal affirmative proposition, A is AB, as follows: All A 
is B, therefore, some B is A; for, if not, no B is A, and, there- 
fore, by conversion, No A is B, but this is inconsistent with 
the given proposition, All A is B. 

734. In the simple conversion of particular affirmative prop- 
ositions, unless the predicate and the subject are co-extensive, 
the w r ord "some" has a different value in the two propositions, 
according to Dr. Bain. He says: "In the couple, 

Some men are dark-haired, 

Some dark-haired beings are men, 
"Some men," as compared with "All men" is a larger fraction 
than some dark-haired beings as compared with all dark-haired 
beings." 

In every true proposition the predicate and subject must be 
co-extensive, because they are merely names for the same 
thing. 

735. Dr. Bain gives an example of the application of this 
process of contraposition to a universal affirmative proposition, 
thus: 

All men are mortal, 
Xo men are immortal, 
Xo immortals are men, 
or, by symbols, thus. 
All X is Y 
Xo X is not-Y 
Xo not-Y is X. 

736. In our system we would state the proposition, All 
men are mortal, by 

A == AB, 
This would cause us to eliminate the combination 
Ab, 
and then we can read the eliminated combination, 



410 



CONVERSION. 



[ Chap. 26. 



No A = b, 

A = no b 
No b = A 

b === no A. 



A 


a 








B 


1 




b 



Fig. 281. 

737. This process of conversion by negation or contrapo- 
sition, is applicable to universal affirmatives. 

"A universal affirmative may be stated as a universal nega- 
tive, thus, 

Every A = B, 

No A = not-B, A is no not-B, 
and this again may be converted into, 

Not-B = not-A, 
for instance, 

Every true poet is a man of genius, which may be converted 
into, 

No true poet is not-a-man-of-genius, 
which may be converted into: 

No one who is not a man of genius is a true poet." 

738. In our system, Every true poet is a man of genius, 
would be stated thus, 

A = AB, 
This would cause us to eliminate Ab, and we can read the 
eliminated combination, 

No b = A, which can be translated, 
No not-genius is a true poet, thus, 



§§ 739-741.] MEANING OF MAY, CAN, ETC. 



411 



A a 

B 

1 b 



Fig. 282. 

The foregoing expression, No not-genius is a true poet, can 
be expressed in the following equivalent phrases, 

"None but a man of genius can be a true poet." 
"A man of genius alone can be a true poet," 
"One cannot be a true poet without being a man of 
genius." 

739. In examples like the above, the words, "may," "can," 
"cannot," etc., have no reference to power exercised by an 
agent; they refer merely to the confidence or doubtfulness we 
feel in respect to some supposition. To say, for instance, that 
"A man who has the plague may recover," does not mean that 
it is in his power to recover if he chooses; but it is only a form 
of stating "Some who have the plague recover." This is an 
ordinary particular affirmative proposition. 

740. So also to say, 

"A virtuous man cannot betray his country," or, 
"It is impossible that a virtuous man should betray his 
country," 
does not mean that he lacks the power to betray his country, 
but it is merely a different way of stating the universal nega- 
tive proposition, 

"No virtuous man betrays his country." 

741. In order to reason correctly, it is necessary, as we have 
remarked before, that propositions should be stated correctly, 
and to do this, we must perceive the exact logical force of the 
given proposition. The correct stating of propositions is of as 



412 CONVERSION. [Chap. 25. 

much importance as the working of them out in the Reasoning 
Frame. 
The preceding examples are taken from Archbishop Whately. 

742. Prof. Bain says, that when we affirm one thing, we 
must be prepared to deny its opposite. A fact can be stated in 
two different ways, thus, 

"This road is level," 
"This road is not inclined." 
These are not two facts, but the same facts stated in two dif- 
ferent ways. This process is named Obversion. 

743. He gives the following examples (the lettering is mine), 
Every man is mortal, every A is B, 

First obvert the predicate, 

Every man is not mortal, every A is not B. 
Next, prefix the sign of negation to the subject, 

No man is immortal, no A is not B. 

744. To obvert the particular affirmative proposition, 
Some men are wise, some A is B, 

first obvert the predicate, and then prefix the sign of negation 
to the predicate. 

Some men are not-wise, i. e., foolish. 

745. To obvert the universal negative proposition, 
No men are gods, no A is B, 

transfer the sign of negation from the subject to the predicate, 
All men are no-gods, 
All A is not-B. 

746. To obvert the particular negative proposition, 
Some men are not wise, 

Some A is not B, 
change the form of the proposition thus, 

Some men are not- wise, i. e., foolish. 
Some A is not-B. 

747. Prof. Bain gives the rule for obversion thus, 
"Obvert the predicate and change the quality of the 

proposition." 



§§ 748, 749.] 



AN EXAMPLE. 



413 



748. Trof. Lotze says, vol. 1, p. 107: 

"We cannot infer from the negation of the universal 
proposition, either the truth or the untruth of the particular.'' 

I think this is a mistake, 

Let A = AB represent the universal proposition ; its nega- 
tion would be represented by, 

No A = B. 

vMake an AB diagram : 



A 


a 




1 




B 






b 



Fig. 283. 

Now, if No A = B, then the combination AB is incon- 
sistent, because it implies that A = B, and we eliminate it by 
making a figure 1 in that section. 

Now, as the combination AB, is non-existent, the untruth of 
the particular proposition AB is AB, which means Some A is 
some B, follows necessarily. 

749. Prof. Lotze says in "Logic" vol. 1, p. 108: "If we deny 
the proposition, 

All S are P, i. e., A is AB, 
the denial is consistent with both the assumptions, E and O, 

No S is P (i. e., No A is B), 

Some S are not P (i. e., Ab = Ab), 
but the second which is included in the first is true in any case; 
consequently the truth of O follows certainly from the untruth 
of A." 



414 



CONVERSION. 



[ Chap. 26. 



By our method we make an AB diagram, then we eliminate 

the AB combination, then we read the AB combination, 

No A = B, this is E, 

and we read the Ab combination, 

All A is some b, thus, 
* 



A 


a 




1 




B 






b 



Fig. 284. 

750. Again, Prof. Lotze says : "If we further deny O, 

Some S are not P (i. e,, Ab = Ab), 
This means, according to what we said above, 

"There is no such thing as Some S which are not P, and 
this is equivalent to A, 

All S are P, i. e., A = AB." 
Make an AB diagram, eliminate the Ab combination: 



A 


a 








B 


1 




b 



Fig. 285. 

And then we can read the AB combination, 
All A is some B. 

751. Again Prof. Lotze says : "If we deny I, this means : 
There is no such thing as Some S which are P, and is 
equivalent to the affirmation of E, 
No S is P. 



g§ 752, 753.] IMPURE CONVERSION. 

Make an AB diagram : 



415 



A 


a 




2 




B 






b 



Fig. 286. 

Let AB = AB represent the proposition I. 

Now if we deny I, we can represent the denial of I by elim- 
inating the AB combination. 

Suppose we eliminate the AB combination by making a fig- 
ure 2 in that section. 

We can read the eliminated combination thus: 
No A = B, 
which is equivalent to the affirmation of E. 

752. Again, Prof. Lotze says, that a when we have a propo- 
sition which can be stated: 

A = B 
and read backward as well as forward, we have a case of what 
is called pure conversion, but when the proposition is of the 
form 

A =AB, 
which means some B, which reads backward, 

Some B = A, 
we have a case of impure conversion. 

It is a very common mistake and also a favorite means of 
deception to convert the proposition, 
A = AB into 
B = A." 

753. Prof. Lotze calls propositions of the form, 

A = B, 
B = A, 
reciprocal judgments. He gives these examples: 



416 CONVERSION. [ Chap. 26. 

"All men are naturally capable of language," 
"All equilateral triangles are equi-angular;" 
They can be converted into, 

"All that is naturally capable of language is man." 
"Every equiangular triangle is an equilateral one." 

754. In speaking of the conversion of the particular affir- 
mative proposition in the old logic, Lotze says : "But when S 
is the genus of which P is a species, as in the proposition, 
"Some dogs are pugs," the only logical admissible conversion, 
"Some pugs are dogs," will contrast unfavorably with the 
actually true one, "All pugs are dogs." The former is no doubt 
true also, but it expresses only a part of the truth which 
appears rather to deny than affirm the other part, that, "All 
other pugs are also dogs." 

We feel this still more if we start with the judgment, "All 
pugs are dogs," and convert it twice over. From the first con- 
version, "Some dogs are pugs," we cannot get back again by the 
second to the original proposition, and thus the logical opera- 
tions have here resulted in eliminating a part of the truth. 
This inconvenience could easily be avoided if the expressions 
of quantity were regarded, as the sense requires that they 
should be as inseparable from their substances, we should then 
formulate the proposition, in the first instance, as follows: 
"All pugs are some dogs;" then by conversion, "Some dogs are 
all pugs," and by a second conversion, "All pugs are some 
dogs." But it is not worth the trouble to improve what are, 
after all, barren formulae." 

755. Speaking of the particular negative judgment: "Some 
S are not P," Prof. Lotze says, "The pure conversion, there- 
fore, "Some P are not S," does not hold good universally, but 
only of those P which are predicates common to different sub- 
jects and are not, therefore, exclusively dependent upon the 
nature of S for their occurrence. For this reason, the propo- 
sition, "Some men are not black," can be converted into, "Some- 
thing black is not man ;" but the judgments, "Some men are not 
pious," "Some are not Christians," would yield "Something 



§§ 756, 757] A NEW TERMINOLOGY. 417 

pious is not man," "Some Christians are not men," both inad- 
missible, because piety and Christianity, though not belonging 
to all men, belong only to men. 

These disadvantages are in general only avoided by joining 
the negative to the predicate, and then converting the proposi- 
tion, "Some S are non-P," like a particular affirmative, into 
"Some non-P are S," e. g., "Something not-black, something 
not-pious, some non-Christians are men. The process necessary 
in this case, has been extended to all judgments under the 
name of conversion by contraposition; in the affirmative judg- 
ments the negation of non-P takes the place of affirmation of P; 
in the negative the affirmation of non-P, takes that of the nega- 
tion of P; the judgments thus changed are then converted 
according to ordinary rules. In this way we get the following 
results: First, for A, "All S are P," "No S is non-P," and so 
non-P is no S;" for I, on the other hand, "Some S are P," the 
transformation into "Some S are non-P," would not, after what 
has been said above, allow any conversion and transposition 
would therefore be impossible; for E again, "No S is P," we get, 
"All S are non-P," "Some non-P are S." 

I would like to say right here, that Prof. Lotze is in some 
respects one of the clearest thinkers in logic that I know of. 

756. Miss Jones, in "Elements of Logic," p. 143, proposes a 
new terminology for the different kinds of conversion. She 
suggests, 

Subversion for subalternation, 
Reversion for simple conversion, 
Intraversion for conversion per accidens, 
Contraversion for contraposition, 
Retroverse for obverted converse, 
Extraversions for added determinants, 
Extraversion for inference by complex conception. 

757. She says that subversion is passing from a com- 
plete proposition to a partial proposition that has the same sub- 
ject, the same predicate, and the same quality, thus: 

27 



418 CONVERSION. [ Chap. 26. 

"Every wind is ill to a broken ship," 
"Some winds are ill to a broken ship." 

758. In obversion, she says that the obverse has the same 
subject and the same quantity as the obvertend, but different 
quality; and the predicate of the obverse is the negative of the 
predicate of the obvertend, thus: 

"All is fine that is fit," 
"Nothing is not fine that is fit." 

The principle of obverting is, that the affirmation (or denial) 
of any predicate, justifies the denial (or affirmation) of its nega- 
tive. 

759. In a reversion, she says that the educt has the predi- 
cate of the educend for its subject, and the subject of the 
educend for its predicate. 

Keverse and Revertend do not differ in quantity or quality; 
only E and I can be reverted. 
To take two simple examples," 

"Those plants are biennials," 

"Some biennials are all those plants," 

"No man is a free agent who cannot command himself," 

"No free agent is a man who cannot command himself." 

760. In intraversion she says, that we infer from an affirma- 
tive proposition, a partial proposition of the same quality, 
which has the predicate of the educend for its subject and the 
subject of the educend for its predicate, thus : 

"An honest miller has a black thumb," 
"Some persons having a black thumb are honest mil- 
lers." 

761. In controversion she says, that the educt differs from 
the educend as follows: 

(1) The subject of the educt is the negative of the predicate 
of the educend. 

(2) The predicate of the educt is the subject of the educend. 

(3) Educt and educend differ in quality. 

(4) Every contraverse, except that of E, has the same quan- 
tity as the contravertend, thus: 



§§762-766.] RETROVERSION. 41 9 

"If is stiff," 
"Not-stiff is not-if." 

762. She says that O cannot be contraverted, because its 
obverse is O, which cannot be converted; and to reach the con- 
traverse of any proposition, it has to be obverted. and then the 
obverse thus obtained has to be converted. 

763. In retroversion she says the educt differs from the 
educend in quality, the predicate of the educend is the subject 
of the educt, and the negative of the subject of the educend 
is the predicate of the educt. 

764. In the universal affirmative proposition only, the educt 
and the educend differ in quantity. She gives these examples: 

"All who love me keep my commandments," 

"Some who keep my commandments are not those who 
do not love me," 

"Some doctrines are universally accepted," 

"Some things universally accepted are not doctrines 
which are not true," 

"Some believers in Spiritualism are these well-known 
writers, 

"These well-known writers are not disbelievers in Spir- 
itualism," 

"These R's are Q's," 

"Some Q's are not-these-R's." 

765. In inversion, she says, "We obtain from a given propo- 
sition a new proposition having the contradictory of the origi- 
nal subject for its subject and the original predicate for its 
predicate. Also the inverse of any proposition differs from 
the invertend in both quality and quantity. A and E are the 
only propositions which can be inverted. The following are 
examples: 



420 CONVERSION. [Chap. 26. 

"No sunshine is without shadow," 



"Some things that are not sunshine are without 

shadow," 
"A friend in need is a friend indeed," 
"Some who are not friends in need are not friends 

indeed." 

766. "Only coincidental propositions can be converted, con- 
traverted, retroverted, or inverted. Only A propositions can 
be intraverted. Adjectivals as well as coincidentals may be 
subverted, obverted and extraverted." 

767. The reader will understand that in this chapter on 
Conversion I am trying to give a brief account of what the 
writers on the old logic have to say in regard to this process. 

768. In our system, as heretofore pointed out, we quantify 
the terms when we state the proposition, so that the subject 
and predicate are equivalent terms. This does away at once 
with the necessity of conversion. The process for making the 
subject and predicate identical is exceedingly simple. If the 
subject is wider than the predicate we reduce it to an equiva- 
lence with the predicate by adding the predicate to the subject. 
Thus, given the propositions, 

Some A = All B 
we state it, 

AB = B 
This means that the A which is B is B, e. g., 
Some dogs are pugs, 
The dogs which are pugs are pugs, or, 
Pug dogs are pugs, 
and of course this proposition can be read backward, 
Pugs are pug dogs. 
If the predicate is wider than the subject, we reduce the 
predicate to the limits of the subject by adding the subject to 
the predicate, thus: 



§ 769.] TABLE OF CONVERSIONS. 421 

If we have the proposition, 

A is some B 
we state it thus, 

A = AB 
This means that A is the B which is A, e. g., 
Men are animals, 

Men are the animals which are men, or, 
Men are animal men, 
which is equally true when read backward, 
Animal men are men. 
I do not think it is possible to reason exactly in complicated 
cases, unless this process of making the terms equivalent is 
pursued. 

769. Dr. Keynes in his work on "Formal Logic'' gives a 
more detailed statement of conversion than any other writer. 
For obtaining a knowledge of the old logic his work is very 
satisfactory. On p. 99 he says, in speaking of the A, E, I, O 
propositions, "We have, therefore, the following table of propo- 
sitions connecting any two terms S and P (the lettering is 
mine): 

A = AB 
B == BA 

No A = B No B = A 
AB = B BA = A 
Ab = b 
Ba = a 
The translations are, 
All A is some B, 
All B is some A, 
No A is B, No B is A, 
Some A is B, Some B is A, 
Some A is not-B, 
Some B is not-A. 
The pair of propositions, A is AB and B is BA, are indepen- 
dent, and the same is true of the pairs, 



422 CONVERSION. 

A not-B is not-B and B not-A is not-A, 
A is AB and B not-A is not-A, 
B is BA and A not-B is not-B." 

Make an AB diagram: 



[ Chap. 26. 



A 


a 






2 


B 


1 




b 



Fig. 287. 

Now, if A = AB, then the combination Ab is inconsistent, 
and we eliminate it by making a figure 1 in that section. 

Again, if B = BA, then the combination Ba is inconsistent 
and we eliminate it by making a figure 2 in that section. 

The Reasoning Frame now shows that the two propositions 
neither concur nor conflict, that is, they are independent. 

770. Make an AB diagram: 



;a 


a 








B 




21 


b 



Fig. 288. 

Now, if Ab = b, then the combination ab is inconsistent 
and we eliminate it by making a figure 1 in that section. 

Again, if Ba == a, then the combination ab is inconsistent 
and we eliminate it by making a figure 2 in that section. 

The Reasoning Frame now shows us that the propositions 
are equivalent, because they both remove the same combina- 



§771.] 



CONTRADICTORIES. 



423 



tion. They are not independent in the sense that the proposi- 
tions A = AB and B = BA are. 

771. Make an AB diagram: 



A 


a 






B 


1 


2 


b 



Fig. 289. 

Now, if A = AB, then the combination Ab is inconsistent 
and we eliminate it by making a figure 1 in that section. 

Again, if Ba = a, then the combination ab is inconsistent 
and we eliminate it by making a figure 2 in that section. 

The Reasoning Frame now shows us that all the b's are 
eliminated. This means that the propositions are contradic- 
tory. 

Make an AB diagram : 



A 


a 






1 


B 




2 


b 



Fig. 290. 

Now, if B = BA, then the combination Ba is inconsistent, 
and we eliminate it by making a figure 1 in that section. 

Again, if Ba = a, then the combination ba is inconsistent, 
and we eliminate it by making a figure 2 in that section. 

The Reasoning Frame now shows that all the a's are elimi- 
nated. This proves that the propositions are contradictory. 



424 



CONVERSION. 



[ Chap. 26. 



772. Dr. Keynes says, that the first pair taken together may 
be called complementary propositions; the second pair, he says, 
are neither coextensive, nor either included within the other 
(?), and they may be called sub-complementary propositions. 

The third pair may be called contra-complementary proposi- 
tions. The fourth pair may also be called contra- complement- 
ary propositions. 

773. Dr. Keynes says that obversion is a process of imme- 
diate inference, in which the inferred proposition, or obverse, 
whilst retaining the original subject, has for its predicate the 
contradictory of the predicate of the original proposition, or 
obvertend. 

774. Make an AB diagram: 



A 


a 








B 


1 




b 



Fig. 291. 



Now, if A = AB ,then the combination Ab is inconsistent, 
and we eliminate it by making a figure 1 in that section. 
We can read the eliminated combination, 

No A = b 
This is the obverse proposition. 



§§ 775, 776] OB VERSION. 

775. Make an AB diagram : 



425 



A 


a 






1 


B 






b 



Fig. 292. 

Now, if AB = B, then the combination Ba is inconsistent 
and we eliminate it by making a figure 1 in that section. 

The Seasoning Frame now shows that the definition of A is, 
A = B | b, 
from which the old logic would infer, 

Some A is B, 
which by the use of a double negative it would convert into, 
Some A is not not-B, 
This is the obverse of No A = B. 

Some A is B. 

776. Make an AB diagram: 



A 


a 




1 




B 






b 



Fig. 293. 

Now, if No A = B, then the combination AB is inconsistent 
and we eliminate it by making a figure 1 in that section. 
The Reasoning Frame now shows us that the definition of A 

is, 

A = Ab. 



426 CONVERSION. 

777. Make an AB diagram : 



[ Chap. 26. 



A a 

B 

1 b 



Fig. 294. 

Now, if Ab = b, then the combination ab is inconsistent and 
we eliminate it by making a figure 1 in that section. 

The Reasoning Fame now shows that the definition of A is, 
A = B | b, 
from which the old logic infers, 

Some A is not-B. 
This is the obverse of 

Some A is not B. 

778. Obversion is called Permutation by Prof. Fowler; 
Equipollence by Prof. Ueberweg; Infinitation by Prof. Bowen; 
Immediate Inference by Privitive Conception by Prof. Jevons; 
Contra version by Prof. DeMorgan, and Contraposition by Prof. 
Spalding. 

779. Dr. Keynes says, "Prof. Bain distinguishes between 
formal obversion and material obversion. By formal obver- 
sion is meant the kind of obversion discussed in the above sec- 
tion, and this is the only kind of obversion that can be properly 
recognized by the formal logician. Material obversion is des- 
cribed as the process of making 'obverse inferences which are 
justified only on an examination of the matter of the propo- 
sition, and the following are given as examples: 

'Warmth is agreeable, therefore, 
'Cold is disagreeable, 
'War is productive of evil, therefore, 
'Peace is productive of good, 



§§ 780-783.] 



CONTRAPOSITION. 



427 



'Knowledge is good, therefore, 

'Ignorance is bad.' 
It is very doubtful if these are legitimate inferences, formal or 
otherwise." 

780. It seems to me improper to call material obversion a 
logical process. In a correct logical process it is impossible to 
get any term in the conclusion that is not given in the premises. 

781. Dr. Keynes says that contraposition is a process of 
immediate inference, in which from a given proposition an- 
other proposition is inferred, having for its subject the contra- 
dictory of the original predicate. 

Thus, given a proposition having A for its subject and B for 
its predicate, we seek to obtain a new proposition having 
not-B for its subject. 

782. Every proposition which admits of contraposition will 
accordingly have two contrapositives, each of which is the 
obverse of the other, for example, in the case of, 

All A = B 
there will be two forms, 
No not-B is A 
All not-B is not-A. 

7S3. Make an AB diagram: 



A 


a 


B 






1 




b 



Fig. 295. 

Now, if A = AB, then the combination Ab is inconsistent 
and we eliminate it by making a figure 1 in that section. 



428 



. CONVERSION. 



[ Chap. 26. 



We can now read the eliminated combination, 
No not-B is A, 
and we can read, 

All not-B is not-A 

784. Dr. Keynes says, that so far as it is necessary to distin- 
guish these forms, we may call that one in which A is the 
predicate, the contrapositive, and the one in which not-A is 
the predicate, the obverted contrapositive. 

785. He gives this rule for obtaining the contrapositive: 
Obvert the original proposition and then convert the proposi- 
tion thus obtained. 

786. Make an AB diagram: 



A 


a 




1 




B 






b 



Fig. 296. 

Now, if No A = B, then the combination AB is inconsistent 
and we eliminate it by making a figure 1 in that section. 

The Keasoning Frame now shows us that we can get the 
following obverse proposition: 

A = Ab 
and the following contrapositive proposition, 

Ab = A, 
which means, 

Some not-B is A. 



§§ 787-789.] 



CONTRAPOSITION. 



429 



787. Make an AB diagram 



A 


a 








B 




1 


b 



Fig. 297. 

Now, if Ab = b, which means, Some A is all not-B, then the 
combination ab is inconsistent and we eliminate it by making 
a figure 1 in that section. 

The Reasoning Frame now shows us that the definition of 
A is, 

A = B | b, 
from which the old logic infers the obverse proposition, 
Some A is not-B. 
We also get the following definition of not-B, 

b = bA, 
which means, 

not-B is some A, 
and this is the contrapositive of, 

Some A is not-B. 

788. Dr. Keynes quotes the following from DeMorgan, 
"Euclid may have been ignorant of the identity of 'Every 

X is Y and every not-Y is not-X,' for anything that appears 
in his writings he makes the one follow the other by a new 
proof each time." 

789. Dr. Keynes says, "In most text books no definition of 
contraposition is given at all, and it may be pointed out that 
in the attempt to generalize from special examples, Jevons, 
in his "Elementary Lessons in Logic," involves himself in diffi- 
culties. For the contrapositive of A he gives, All not-A is 
not-B; O, he says, has no contrapositive (but only a converse 



430 



CONVERSION. 



[ Chap. 26. 



by negation, Some not-B is A); and for the contrapositive of 
E he gives, No B is A. 

It is impossible to discover any definition of contraposition 
that can yield these results. Assuming that in contraposition 
the quality of the proposition is to remain unchanged, as in 
Jevons' contrapositive of A, then the contrapositive of both E 
and O is, "Some not-B is not not-A." (For S and P I have sub- 
stituted A and B.) 

790. Dr. Keynes says that Inversion is a process in which 
from a given proposition another proposition is inferred, hav- 
ing for its subject the contradictory of the original subject. 

Given a proposition with A for subject and B for predicate, 
we obtain by inversion a new proposition with not-A for sub- 
ject. The original proposition is called the invertend and the 
inferred proposition the inverse. 

791. The predicate may be either B or not-B. The former is 
called the inverse and the latter the obverted inverse. 

792. Make an AB diagram : 



A 


a 








B 


1 




b 



Fig. 298. 

Now, if A = AB, then the combination Ab is inconsistent 
and we eliminate it by making a figure 1 in that section. 

The Seasoning Frame now shows us that the definition of 
a is, 

a = B | b, 
from which the old logic infers, 

Some not-A is not-B. 



5§ 793-795.] OBVERSION. 

793. Make an AB diagram: 



431 



A 


a 




1 




B 






b 



Fig. 299. 

Now, if No A = B, then the combination AB is inconsistent 
and we eliminate it by making a figure 1 in that section. 

The Reasoning Frame now shows us that the definition of 
not- A is, 

a = B | b, 
from which the old logic infers, 

Some not-A is B. 

794. This is the way in which Dr. Keynes proceeds with 
the proposition, 

All A is B, 
therefore (by obversion) No A is not-B, 
therefore (by conversion) No not-B is A, 
therefore (by obversion) All not-B is not-A, 
therefore (by conversion) Some not-A is not-B, 
therefore (by obversion) Some not A is not B, 
Which last is the desired form. 

795. Make an AB diagram: 



A 


a 






1 


B 






b 



Fig. 300. 



432 



CONVERSION. 



[ Chap, 26. 



Now, if AB == B, which means, Some A is all B, then the 
combination aB is inconsistent and we eliminate it by making 
a figure 1 in that section. 

The Keasoning Frame now shows us that the definition of 
not-A is, 

a = ab, 
which means, 

All not-A is Some not-B. 
and this is the inverse of the proposition, 

Some A is B. 
Now, if Ab = b, which means, Some A is all not B, then 
the combination ab, is inconsistent and we eliminate it by 
making a figure 1 in that section. 

Make an AB diagram : 



A 


a 








B 




1 


b 



Fig. 301. 

The Keasoning Frame now shows us that the definition of 
not-A is, 

a = aB, 
which means, 

All not-A is some B, 
and this is the inverse of the proposition, 

Some A is not B. 

796. Dr. Keynes says, "We may now inquire further in 
what cases it is possible to infer a proposition with not-A as 
subject," and he answers it, " The required proposition can be 
obtained only if the given proposition is universal." 



§ 797.] INVERSION. 433 

797. Dr. Kernes says, "In passing from All A is B, to its 
inverse, Some not A is not B, there is an apparent illicit pro- 
cess, which is far from easy either to account for or 
explain away. For the term B, which is undistributed in the 
premise, is distributed in the conclusion, and yet, if the univer- 
sal validity of obversion and conversion is granted, it is impos- 
sible to detect any flaw in the argument by which the conclu- 
sion is reached. On this ground Prof. Ray rejects the validity 
of the above inference. "If a term is not distributed in the 
premise, it cannot be distributed in the conclusion, that is, if 
a term is taken in the premise to mean at least one thing 
denoted by it, it cannot in the conclusion be taken to mean, all 
things denoted by it. The above conclusion is, therefore, 
inadmissible. It is obtained from the original premise by the 
processes of obversion and conversion, and the fallacy lies not 
in the process of conversion, but in that of obversion, which 
assumes that the term B has a contradictory and is therefore 
limited in its sphere, although in the original premise, its limi- 
tation is not implied and it may cover the whole sphere of 
thought and existence." ("Deductive Logic, p. 313.") 

"Instead, however, of thus denying altogether the valid- 
ity of the inference under consideration, it is better 
to investigate the conditions of its validity. There 
can be no question that it is valid under the exis- 
tential assumption upon which we have been proceeding. 
By the aid of diagrams this can be shown directly and without 
the intervention of the processes of obversion and conversion. 
We must then deny that, (under our present assumption) any 
illicit process, whatever, is involved in the inference. In other 
words, admitting contradictory terms, and assuming that the 
original terms and their contradictories are all represented in 
the universe of discourse, it is not correct to say that a term 
not distributed in the premise of an immediate inference, may 
never be distributed in the conclusion. For, although a terin 
(B), may be undistributed relatively, (A), it may nevertheless 
be distributed relatively to the contradictory of A. When we 
say All A is B, B is undistributed relatively to A, but implic- 
28 



434 



CONVERSION. 



[ Chap. 26. 



itly it is at the same time, entirely excluded from some portion 
of not-A, and is, therefore, distributed relatively to that portion 
of not-A." 

798. Make an AB diagram: 



A 


a 








B 


1 




b 



Fig. 302. 

Let us take the proposition, 

All A = B, which we can state thus: 
A = AB. 
Now, if A = AB, then the combination Ab is inconsistent 
and we eliminate it by making a figure 1 in that section. 
Reading the eliminated combination we get, 

No A = b. 
This is the obverse. 
We can also read, 

B — A | a, 
which the old logic would translate, 

Some B is A. 
This is the converse. 
We can also read, according to the old logic, 

Some B is not not-A, 
This is the obverJ:ed converse. 
We can also read the eliminated combination, 

No b = A 
This is the contrapositive. 
We can also read the combination, 
All b = a 
This is the obverted contrapositive. 



§3 799, 800.] 



NEW READINGS, 



486 



We can also read, 

a = B | b 
which the old logic would translate, 

Some a = b 
This is the inverse and the obverted inverse is the same. 

799. The above is a list of the inferences which Dr. Keynes 
gives, but I do not think that he has exhausted the list accord- 
ing to the methods of the old logic, for instance, we can read 
according to the old logic, 

Some B is not-A 

Some not-B is no A 

Some not-A is B, 
and by the use of double negatives we can read, according 
to the old logic, 

Some not-A is not not-B 

Some A is no not-B 

800. Make an AB diagram: 



A 


a 




1 




B 






b 



Fig. 303. 
Let us take the proposition, 

No A = B. 
Now, if No A = B, then the combination AB is inconsistent 
and we eliminate it by making a figure 1 in that section. 
AYe can read, 

A = Ab, 
which means, 

All A is not-B 
This is the obverse. 
We can also read, 

No B = A, 
This is the converse. 



436 CONVERSION. [ Chap. 26. 

We can also read, 

B == Ba, 
which means, 

All B is not- A, 
This is the obverted converse. 
We can also read, 

b = A | a, 
which means, according to the old logic, 
Some not-B is A, 
This is the contrapositive. 
W T e can also read, 

Some not-B is not not-A, 
This is the obverted contrapositive. 
We can also read, 

a = B | b, 
which means, according to the old logic, 
Some not-A is B, 
This is the inverse. 
We can also read, 

Some not-A is not not-B, 
This is the obverted inverse. 

801. The above are the inferences given by Dr. Keynes, 
but, according to the old logic, we can also read, it seems to 
me, 

Some not-B is not A, 
Some A is no B 
Some not A is not-B, 
Some B is no A. 

802. Let us take the proposition, 
Some dogs are all pugs. 

Let A = dogs, 
B = pugs. 
We can state the proposition thus: 

BA == B. 



§ 803.] AN EXAMPLE. 

Make an AB diagram: 



437 



A 


a 






1 


B 






b 



Fig. 304. 

Now, if B = BA, then the combination Ba is inconsistent 
and we eliminate it by making a figure 1 in that section. 

We can now make, according to the old logic, the following 
readings: 

(1) Some A is B, i. e., Some dogs are pugs, 

(2) Some A is not-B, i. e., Some dogs are not-pugs, 

(3) Some A is no not-B, i. e., Some dogs are no not-pugs, 

(4) No not-A is B, i .e., No not-dogs are pugs, 

(5) No B is not-A, i. e., No pugs are not-dogs, 

(G) All not-A is some not-B, i. e., All not-dogs are some not- 
pugs, 

(7) All B is some A, i. e., All pugs are some dogs, 

(8) Some not-B is some A, i. e., Some not-pugs are some dogs, 

(9) Some not-B is some not-A, i. e., Some not-pugs are some 

not-dogs, 

(10) Some a is no B, i. e., Some not-dogs are no pugs. 

(11) Some B is no a, i. e., Some pugs are no not-dogs. 

803. Let us take the proposition, 
Salt is chloride of sodium, 
Let A = salt, 

B = chloride of sodium. 
The proposition can be stated thus: 

(1) A = B 

(2) B = A 



^38 CONVERSION. 

Make an AB diagram: 



[ Chap. 26. 



A 


a 






2 


B 


1 




b 



Fig. 305. 

Now, if A = B, then the combination Ab, is inconsistent 
and we eliminate it by making a figure 1 in that section. 

Again, if B = A, then the combination Ba is inconsistent 
and we eliminate it by making a figure 2 in that section. 

We can make the following readings: 

(1) All A = all B, 

All salt is all chloride of sodium, 

(2) All B == All A, 

All chloride of sodium is all salt, 

(3) All a = all b, 

All not-salt is all not-chloride of sodium, 

(4) All b = all a, 

All not-chloride of sodium is all not-salt, 

(5) All A = no b, 

All salt is no not-chloride of sodium, 

(6) All B = no a, 

All chloride of sodium is no not-salt, 

(7) All b = no A, 

All not-chloride of sodium is no salt, 

(8) All a = no B, 

All not-salt is no chloride of sodium. 

804. We will now give some examples taken from Arch- 
bishop Whately's "Elements of Reasoning," pp. 118, 119 and 
120, to exhibit the application of conversion to the syllogism. 

All wits are dreaded, 

All wits are admired, 



§ 804.] 



AN EXAMPLE. 



439 



Some who are admired are dreaded. 

We can state the propositions thus: 
A = AB 
A = AC 
CB = CB 

Make an ABC diagram: 



AB 


Ab 


aB 


ab 




2 


1 






C 


12 






c 



Fig. 306. 

Now, if A = AB, then the combinations containing Ab are 
inconsistent and we eliminate them by making a figure 1 in 
those sections. 

Again, if A == AC, then the combinations containing Ac are 
inconsistent and we eliminate them by making a figure 2 in 
those sections. 

The given propositions were reduced into Darii, by convert- 
ing by limitation the minor premise. 

All wits are dreaded, 

Borne who are admired are wits. 

Some who are admired are dreaded, 
which we can state, 

A = AB 
CA = A 
CB = CB. 



440 CONVERSION. 

Make an ABC diagram: 



[Chap. 26. 



AB 


Ab 


aB 


ab 






1 






C 


2 


21 






c 



Fig. 307. 

Now, if A = AB, then the combinations containing Ab are 
inconsistent and we eliminate them by making a figure 1 in 
those sections. 

Again, if A = CA, then the combinations containing cA 
are inconsistent and we eliminate them by making a figure 2 
in those sections. 

The appearance of the two Reasoning Frames proves the val- 
idity of the conversion. 

The original example was in Darapti. 

805. Let us take these examples in Camestres: 

All true philosophers account virtue a good in itself, 

The advocates of pleasure do not account, etc., 

Therefore they are not true philosophers, 
which we can state, 

A = AB 
= Cb 
C = Ca 

Make an ABC diagram: 



AB 


Ab 


aB 


ab 




2 


1 


2 









1 






c 



Fig. 308. 



§606] 



CAMESTRES. 



441 



Now, if A = AB, then the combinations containing Ab are 
inconsistent and we eliminate them by making a figure 1 in 
those sections. 

Again, if = Cb, then the combinations containing CB are 
inconsistent and we eliminate them by making a figure 2 in 
those sections. 

This syllogism was reduced to Celerant by simply converting 
the minor, and then transposing the premises. 

Those who account virtue a good in itself are not advo- 
cates of pleasure, 
All true philosophers account virtue, etc, therefore, 
No true philosophers are advocates of pleasure, 
which we can state, 

B = Be 
A = AB 
NoA= C 

Make an ABC diagram: 



AB 
1 


Ab 


aB 


ab 




2 


1 




C 
c 


2 







Fig. 309. 

Now. if B = Be, then the combinations containing BC are 
inconsistent and we eliminate them by making a figure 1 in 
those sections. 

Again, if A = AB, then the combinations containing Ab are 
Inconsistent and we eliminate them by making a figure 2 in 
those sections. 

The appearance of the two Reasoning Frames proves the val- 
idity of the conversion. 

806. Let us take this syllogism in Baroco: 

Every true patriot is a friend to religion, 



442 



C0NVER-I3N. 



[ Chap. 26. 



Some great statesmen are not friends to religion, therp 

fore, 
Some great statesmen are not true patriots. 
We can state it thus: 

A == AB 
Cb == Cb. 
Make an ABC diagram: , 



AB 


Ab 


aB 


ab 






1 







C 




1 




c 



Fig. 310. 

Now, if A = AB, then the combinations containing Ab are 
inconsistent and we eliminate them by making a figure 1 iu 
those sections. 

This syllogism was converted into Ferio by converting the 
major premise by negation, i. e., contraposition. 

He who is not a friend to religion is not a true patriot, 
Some great statesmen are not friends to religion. 
They can be stated thus : 

b = ba 
Cb = Cb. 

Make an ABC diagram : 



AB 


Ab 


aB 


ab 






1 






C 




1 






c 



Fisr. 311. 



3 807.] 



BOCARDO. 



443 



Now, if b = ba, then the combinations containing Ab are 
inconsistent and we eliminate them by making a figure 1 in 
those sections. 

The appearance of the two Reasoning Frames proves the 
validity of the conversion. 

807.Let us take this syllogism in Bocardo: 
Some slaves are not discontented, 
All slaves are wronged, therefore, 
Some who are wronged are not discontented. 
The premises may be stated thus: 
Ab = Ab 
A = AC 
Make an ABC diagram: 



AB 


Ab 


aB 


ab 










C 


1 


1 






c 



Fig. 312. 

Ab = Ab will not cause us to eliminate any combinations. 
Now, if A = AC, then the combinations containing Ac are 
inconsistent and we eliminate them by making a figure 1 iu 
those sections. 

The major premise was converted by negation (contrapo- 
sition) and the premises were transposed. 
All slaves are wronged, 

Some who are not discontented are slaves, therefore, 
Some who are not discontented are wronged, therefore, 
Some who are wronged are not discontented. 
The premises may be stated thus: 
A = AC 
bA = bA 



444 

Therefore, 

Therefore, 

Make an ABO diagram : 



CONVERSION. 



[ Chap. 26. 



bC == bC, 
Cb = Cb. 



AB 


Ab 


aB 


ab 












C 


1 


1 






c 



Fig. 313. 

Now, if A = AC, then the combinations containing Ac are 
inconsistent and we eliminate them by making a tigure 1 in 
those sections. 

bA = bA will not cause us to make any elimination. 
The appearance of the two Reasoning Frames proves the 
validity of the conversion. 

808. The old logic had another method called Reductio ad 
impossible. The following is an example: 

All true patriots are friends to religion, 
Some great statesmen are not friends to religion, there- 
fore, Some great statesmen are not true patriots. 
We can state the premises thus : 
A == AB 
Cb = Cb, therefore, 
Ca = Ca. 



§ 808.] AN EXAMPLE. 

Make an ABC diagram: 



445 



AB 


Ab 


aB 


ab 






1 






C 
c 




1 







Fig. 314. 

Now, if A = AB, then the combinations containing Ab are 
inconsistent and we eliminate them by making a figure 1 in 
those sections. 

If the conclusion Ca = Ca be not true, then its contradic- 
tory must be true, viz.: 
All great statesmen are true patriots. 
This can be stated thus: 

C = CA, 
Substitute this for the minor premise and the syllogism 
will now stand thus: 

All true patriots are friends to religion, 
All great statesmen are true patriots, therefore, 
All great statesmen are friends to religion. 
They can be stated thus: 

A = AB 
C = CA, 
Therefore, 

C = CB. 
Make an ABC diagram: 



AB 


Ab 


aB 


ab 






1 


2 


2 


C 




1 






c 



Fig. 315. 



446 CONVERSION. [ Chap. 26. 

Now, if A = AB, then the combinations containing Ab are 
inconsistent and we eliminate them by making a figure 1 in 
those sections. 

Again, if C = CA, then the combinations containing Ca are 
inconsistent and we eliminate them by making a figure 2 in 
those sections. 

The Reasoning Frame now shows that C = CB. 
All great statesmen are friends to religion. 

But this conclusion is the contradictory of the original 
minor premise, therefore, it must be false, because the premises 
are always taken to be true. Therefore, one of the premises 
by which the conclusion has been correctly proved, must be 
false also, but, as the major premise is true, the falsity must 
be in the new minor premise, which is the contradictory of the 
original conclusion. 

Therefore, the original conclusion must be true. This kind 
of reasoning was employed for Baroco and Bocardo in the old 
logic. 

809. Miss Jones says in "Elements of Logic," p. 147, "With 
Inferential (Hypotheticals), there seems to be only one kind 
of Eversion which corresponds pretty nearly to Contraversion 
e. g., from any hypothetical. 

If A, then C, 

Another hypothetical, 

If not C then not A, 
may be educed. 

If A then C means, 

If A = B then = D. 

If not C then not A, means, 

If C = d then A = b. 



810, 811.] HYPOTHETICALS. 

Make an A BCD diagram : 



447 



AB 


Ab 


aB 


ab 












CD 


21 




2 


2 


Cd 


1 








cD 


1 








cd 



Fig. 31G. 

The propositions can be stated thus: 

(1) AB = ABCD 

(2) Cd = CdAb 

Now, if AB = CD, then the combinations containing 
ABCd, ABc are inconsistent and we eliminate them by mak- 
ing a figure 1 in those sections. 

Again, if Cd = Ab, then the combinations containing 
ABCd, aCd, are inconsistent and we eliminate them by mak- 
ing a figure 2 in those sections. 

The result shows that the proposition, 

If not C then not A is consistent with the proposition, if A 
then C, but that it is not an inference from it. 

810. Again Miss Jones says. "For instance, the conditional, 

(1) If any flower is a Datura that flower is fragrant, 
is equivalent to the categorical, 

(2) Any flower that is Datura is fragrant. 

Let A = flower, 
B = Datura, 
C = fragrant. 
The premises can be stated thus: 

(1) AB = ABC 

(2) AB = ABC 

So that Miss Jones' statement is correct. 

811. Let us take this example : 

(1) If honesty is not the best policy, life is not worth living, 

(2) Life is not worth living or honesty is the best policy. 



448 



CONVERSION. 



[ Chap. 26. 



Let A = honesty, 
B = best policy, 
C = life, 
D = worth living. 
The propositions can be stated thus: 

(1) Ab = AbCd 

(2) C = Cd | A = AB 
Make an A BCD diagram: 



AB 


Ab 


aB 


ab 






213 


2 


2 


CD 


32 








Cd 




31 






cD 




31 






cd 



Fig. 317. 

Now, if Ab = Cd, then the combinations containing AbCD, 
Abe, are inconsistent and we eliminate them by making a 
figure 1 in those sections. 

Again, if C = Cd, except where A = AB, then the combi- 
nations containing ABCd, AbCD, aCD, are inconsistent and 
we eliminate them by making a figure 2 in those sections. 

Again, if A = AB, except where C = Cd, then the combi- 
nations containing ABCd, AbCD, Abe, are inconsistent and 
we eliminate them by making a figure 3 in those sections. 

The Reasoning Frame now shows, 

(1) That the two propositions are not equivalent. 

(2) That they are consistent. 

(3) That we can infer (1) from (2). 

(4) We cannot infer (2) from (1). 



CHAPTER XXVII. 



THE ELIMINATION OF NEGATIVE TERMS. 



812. Dr. Keynes says in "Formal Logic." p. 113, "The pro- 
cess of obversion enables us by the aid of negative terms, to 
reduce all propositions to the affirmative form * * * It 
is of course clear that by means of obversion, we can get rid 
of a negative term occurring as the predicate of a proposition. 
The problem is more difficult when the negative term occurs 
as subject, but in this case, elimination may still be possible." 

813. Dr. Keynes says, "We may even be able to get rid of 
two negative terms; for example, 

All not A is some not B, 
is equivalent to, 

All B is some A. 
The two propositions may be stated thus: 

a = ab 
B = BA 
Make an AB diagram: 



A 


a 






12 


B 






b 



Fig. 318. 

Now, if a = ab, then the combination aB is inconsistent 
and we eliminate it by making a figure 1 in that section. 

Again, if B = BA, then the combination Ba is inconsistent 
and we eliminate it by making a figure 2 in that section. 

The result proves that the two propositions are equivalent, 

29 



450 



ELIMINATION OF NEGATIVE TERMS. 



[ Chap. 27. 



because they both cause the elimination of the same combi- 
nation. I 

814. Again Dr. Keynes says, p. 114, "No not-A is not-B is 
equivalent to the statement that Nothing is both riot-A and 
not-B, and this becomes by obversion, Everything is either 
A or B." 

The propositions may be stated thus: 

(1) No a = b 

(2) Everything is A | B. 
Make an AB diagram : 



A 


a 








B 




1 


b 



Fig. 319. 

Now, if No a = b, then the combination ab is inconsistent 
and we eliminate it by making a figure 1 in that section. 
Make another AB diagram : 



A 


a 




1 




B 




1 


b 



Fig. 320. 

Now, if Everything, (i. e., every combination) = A I B, then 
the combinations AB, ab, are inconsistent and we eliminate 
them by making a figure 1 in those sections. We assume that 
or is exclusive. 

An examination of the two Reasoning Frames shows us that 
the two propositions are not equivalent. 



§§815, 816] 



EQUIVALENTS. 



451 



815. Let us take this example, 

Are the following propositions equivalents? 

(1) All A is some B 

(2) All not-B is some not-A 

(3) Nothing is A not-B 

(4) Everything is not-A | AB 
The propositions may be stated thus: 

(1) A = AB 

(2) b = ba 

(3) No A = b 

(4) Everything = a | AB. 
Make an AB diagram: 



A 


a 








B 


2 1 
43 




b 



Fig. 321. 

Now, if A = AB, then the combination Ab, is inconsistent 
and we eliminate it by making a figure 1 in that section. 

Again, if b = ba, then the combination Ab is inconsistent 
and we eliminate it by making a figure 2 in that section. 

Again, if No A = b, then the combination Ab, is inconsist- 
ent and we eliminate it by making a figure 3 in that section. 

Again, if Everything (i. e., every combination), == a | AB, 
then the combination Ab is inconsistent and we eliminate it by 
making a figure 4 in that section. 

The result proves that the four propositions are equivalents. 

816. Let us take this example, 

Are the following propos'tions equivalents? 

(1) All not-A is some not-B 

(2) All B is some A 

(3) Nothing is not-A, B 

(4) Everything is A or not-A, not-B. 



452 



ELIMINATION OF NEGATIVE TERMS. 



[ Chap. 27. 



The propositions may be stated thus: 

(1) a = ab 

(2) B = BA 

(3) No a = B 

(4) Everything = A 
Make an AB diagram : 



ab. 



A 


a 






21 
4 3 


B 






b 



Fig. 322. 

Now, if a = ab, then the combination aB is inconsistent 
and we eliminate it by making a figure 1 in that section. 

Again, if B = BA, then the combination aB is inconsistent 
and we eliminate it by making a figure 2 in that section. 

Again, if No a = B, then the combination aB is inconsist- 
ent and we eliminate it by making a figure 3 in that section. 

Again, if Everything (i. e., every combination), === A | ab, 
then the combination aB is inconsistent and we eliminate it by 
making a figure 4 in that section. 

The result now proves that the given propositions are equiv- 
alent. 

817. Let us take this example, 

Are the following propositions equivalents? 

(1) All A is some not-B 

(2) All B is some not-A 

(3) Nothing is AB 

(4) Everything is not-A or A not-B. 
The propositions may be stated thus: 

(1) A = Ab 

(2) B = Ba 

(3) No A = B 

(4) Everything = a | Ab. 



818.] EQUIVALENTS. 

Make an AB diagram: 



453 



A 


a 




21 
43 




B 






b 



Fig. 323. 

Now, if A = Ab, then the combination AB is inconsistent 
and we eliminate it by making a figure 1 in that section. 

Again, if B = Ba, then the combination AB is inconsistent 
and we eliminate it by making a figure 2 in that section. 

Again, if No A = B, then the combination AB is inconsist- 
ent and we eliminate it by making a figure 3 in that section. 

Again, if Everything (i. e., every combination), = a | Ab, 
then the combination AB is inconsistent and we eliminate it by 
making a figure 4 in that section. 

The result proves that the propositions are^equivalant. 

818. Let us take this example, 

(1) All not-A is some B 

(2) All not-B is some A 

(3) Nothing is not-A, not-B. 

(4) Everything is A or not-A, B. 
The propositions may be stated thus: 

(1) a = aB 

(2) b = bA 

(3) No a = b 

(4) Everything = A | aB 



454 ELIMINATION OF NEGATIVE TERMS. [ Chap. 27. 

Make an AB diagram : 



A 


a 


B 






21 
43 


b 



Fig. 324. 

Now, if a = aB, then the combination ab is inconsistent, and 
we eliminate it by making a figure 1 in that section. 

Again, if b = bA, then the combination ab is inconsistent, 
and we eliminate it by making a figure 2 in that section. 

Again, if No a = b, then the combination ab is inconsistent, 
and we eliminate it by making a figure 3 in that section. 

Again, if Everything (i. e. every combination) = A | aB, then 
the combination ab is inconsistent, and we eliminate it by 
making a figure 4 in that section. 

The Reasoning Frame now shows that the given proposi- 
tions are equivalent. 

819. Let us take this example : 

Are the following propositions equivalents: 

(1) Every A is B 

(2) Every not-B is not- A 

(3) Nothing (i. e., no combination) is both A and 
not-B 

(4) Everything (i. e., every combination) is either B 
or not- A, not-B. 

The propositions can be stated thus: 

(1) A = AB 

(2) b = ba 

(3) No A = b 

(4) Everything = B | ab 



§ 820.] NEGATIVE EQUIVALENTS. 

Make an AB diagram : 



455 



A 


a 


B 
b 






21 
43 





Fig. 325. 

Now, if A = AB, then the combination Ab is inconsistent, 
and we eliminate it by making a figure 1 in that section. 

Again, if b = ba, then the combination Ab is inconsistent, 
and we eliminate it by making a figure 2 in that section. 

Again, if No A = b, then the combination Ab is inconsistent, 
and we eliminate it by making a figure 3 in that section. 

Again, if Everything = B | ab, then the combination Ab is 
inconsistent, and we eliminate it by making a figure 4 in that 
section. 

The Reasoning Frame now shows that the given proposi- 
tions are equivalents. 

820. This diagram illustrates the fundamental laws of 
thought. 

The propositions (1) and (2) are opposites. If one is true the 
other is true. Each is a necessary inference from the other, 
and each is the equivalent of the other. 

We might call them the two extremes. 

If either one is true, then every proposition composed of half 
of one and half of the other would be false, thus: 

(1) Every A is not-B is false 

(2) Every not-B is A is false 

(3) Every not-A is B is false 

(4) Every B is not-A is false 

Whenever we have any proposition given to us, if we will 
find its negative equivalent, then we can tell at once that any 
proposition which is composed partly of the given proposition 
and partly of the opposite is false. 



456 



ELIMINATION OF NEGATIVE TERMS. [Chap. 27. 



821. Dr. Keynes relates that one of the old Greek logi- 
cians, Alexander of Aphrodisias, established the conversion of 
E by means of a syllogism in Ferio. 

No A is B 
therefore, 

No B is A 
for, if not, then by the law of contradiction 

Some B is A 
and we have this syllogism, 

No A is B 

Some B is A 
therefore, 

Some B is not B 
A reductio ad absurdum. 
The premises can be stated thus : 

(1) No A = B 

(2) BA = A 
Make an AB diagram : 



A a 

1 B 

2 b 



Fig. 326. 

Now, if No A = B, then the combination AB, is inconsis- 
tent and we eliminate it by making a figure 1 in that section. 

Again, if A = BA, then the combination Ab is inconsistent 
and we eliminate it by making a figure 2 in that section. 

The Reasoning Frame now shows that all the A's are elimin- 
ated and that the premises are inconsistent. The note at the 
foot of the page says, "The conversion of A and the conversion 
•of I may be established similarly." 



§ 822.] 



CONTRAPOSITION. 



457 



822. Dr. Keynes says on page 121, of Formal Logic, 
''The contraposition of A may also be established indirectly 
by means of a syllogism in Darii." 
All A is B 
Therefore, 

No not-B is A 
for, if not, 

Some not-B is A 
and we have the following syllogism, 

(1) All A is B 

(2) Some not-B is A 
therefore, 

(3) Some not-B is B 
which is absurd. 

Make an AB diagram: 



A 


a 




2 




B 


1 




b 



Fig. 327. 

The premises can be stated : 

(1) A = AB 

(2) bA = A. 

Now, if A = AB, then the combination Ab is inconsistent 
and we eliminate it by making a figure 1 in that section. 

Again, if A = bA, then the combination AB is inconsistent 
and we eliminate it by making a figure 2 in that section. 

The Reasoning Frame naw shows us that all the A's are 
eliminated. This. proves that the premises are inconsistent, 
i. e., absurd. 



CHAPTER XXVIII. 

LOGICAL EXISTENCE. 

823. Some writers on the old logic have a good deal to say 
on the question whether the existence of A or of B is neces- 
sarily implied by their use in a proposition. 

We may suppose, says Dr. Keynes: 

(1) That every proposition implies the existence of both sub- 
ject and predicate and their contradictories. 

(2) That every proposition implies simply the existence of its 
subject. 

(3) That no proposition implies the existence of its subject 
or of its predicate. 

(4) That particulars imply the existence of their subjects 
and universals do not. 

824. It is a fundamental proposition in this system, that 
subject and predicate are names for the same thing, conse- 
quently, both subject and predicate and their contradictories, 
imply the existence in thought of every object of which they 
are names. We believe that negative terms are names just the 
same as affirmative terms are. 

All A is all B, means that the same thing has the names A 
and B. It also means that there are no things called A, which 
are also called a. 

The proposition No A = B, means that the things which are 
called A, are also called b. 

The proposition Some A = B, means that there are some 
things which are called A and B. 

The proposition Some A = b, means that there are some 
things which have the names of A and b. 

825. As to whether the terms used in logic imply the exist- 
ence of the objects of which they are names, outside of thought, 
is a question with which logic has nothing to do. Logic is only 



§§826-830] THE UNIVERSE OF DISCOURSE. 459 

concerned with words and with the thoughts which the words 
represent, that is, with words and their meanings. The logi- 
cian must understand the meanings of the terms which are 
used in the proposition given to him. His object, then, is to 
find out all the equivalent meanings of the given proposition. 

826. I agree with Prof. Jevons, that in deductive logic there 
cannot be any question about existence. Words and their 
meanings alone, concern the logician. He has no interest in 
the things which they represent. 

827. We have seen, in working out our examples, that when- 
ever any letter-term, affirmative or negative, was totally elim- 
inated, that we could draw no conclusions. This, it seems to 
me, tends to prove that the existence in the Universe of Dis- 
course of every term used in the given proposition, is absolutely 
necessary to correct reasoning. 

828. No predication, it seems to me, can be made about 
things which do not exist. Even contradictory propositions 
must be made about things which exist in thought. 

829. In our system, the Universe of Discourse is repre- 
sented by a square, which is divided into a certain number of 
sections, determined by the number of terms and their contra- 
dictories. Our Universe of Discourse is limited to the terms 
given us. Our propositions are limited to a given Universe of 
Discourse. In that Universe of Discourse they are either con- 
sistent or inconsistent. Outside of that Universe of Discourse 
we do not know what they are. 

Under other circumstances, that is, in another Universe of 
Discourse, propositions which were once consistent, may now 
be inconsistent, not because of any change in existence, but 
because of a change in names. 

Suppose I say, there are no such things as unicorns. 
Whether unicorns have an actual material existence, is not 
a logical question; it is a question of fact which science must 
answer. 

830. In my judgment, logic has no more to do with actual 
existences in solving its problems, than arithmetic has. If a 



460 LOGICAL EXISTENCE. [ Chap. 28. 

boy were asked: How much nine unicorns would come to at 
nine dollars apiece, how utterly absurd it would be for him 
to insist on knowing before he proceeded to solve the problem 
whether there were any real, material unicorns. 

Take this example: "No person condemned for witchcraft, 
in the reign of Queen Anne, was executed." This means: Every 
person condemned for witchcraft, in the reign of Queen Anne, 
was not executed, and any person who was executed in the 
reign of Queen Anne, was not condemned for witchcraft. 

Both of these propositions imply the existence in thought of 
persons who have been condemned for witchcraft and of per- 
sons who have been executed. Neither of them is a proposition 
about a non-existing subject. 

831. Dr. Keyne's position seems to me to be that particulars 
imply the actual existence of their subjects, and universals do 
not. His tables of the equivalences of propositions, seem 
to be based on that theory. 

832. Miss Jones makes a very good point when she says: 
"A further point is, that unless the very positing of a term sig- 
nifies the existence of something named by the term, we could 
never say, S is P, since the mere symbol S is certainly not the 
symbol P." 



CHAPTER XXIX. 

NUMERICAL REASONING. 

833. I have already said that I considered numerical reason- 
ing as distinct from logical reasoning. The time may come, 
however, when they will be unified by the discovery of a system 
which will enable us to state logical propositions in numbers, 
so that we can proceed to solve logical problems in the same 
way in which we now solve arithmetical problems, or, by the 
discovery of a logical system which will enable us to state 
arithmetical problems in logical s} T mbols and solve arithmeti- 
cal problems by means of Reasoning Frames, or some other 
logical invention, but at present I consider it useless to try 
to solve numerical problems by the processes of Formal Logic. 
Logic is the explanation of the meanings of names; arithmetic 
deals with the properties of numbers; numbers are not merely 
names, though a number may be used for a name, and when 
so used logic can interpret its meanings. 

834. We may, however, consider some of the problems in 
numerical logic which have engaged the attention of other 
writers on logic, for the purpose of ascertaining wmat success 
they have met with. On page 333, Formal Logic, Prof. Keynes 
gives this example of numerically definite reasoning: 

<k If 70 per cent of M are P, and 60 per cent are S, then at 
least 30 per cent are both S and P." 

The argument may be put as follows: On the average, of 
100 M's, 70 are P and 60 are S. Suppose that the 30 M's which 
are not P are S, still, 30 S's are to be found in the remaining 
70 M's which are P's; and this is the desired conclusion." 

Of course this reasoning is correct but I am unable to demon- 
strate its validity by the Reasoning Frame. 

835. Dr. Keynes on p. 332, gives the following example 
of valid reasoning: 



462 



NUMERICAL REASONING. 



[ Chap. 29. 



Most M is P 
Most M is S 
Therefore, Some S is P. 
Let A == Most M 
B = P 
C = S. 
The propositions can be stated thus: 
A = AB 
A = AC 
Make an ABC diagram : 



AB 


Ab 


aB 


ab 






1 






C 


2 


12 






c 



Fig. 328. 

Now, if A = AB, then the combinations containing Ab are 
inconsistent and we eliminate them by making a figure 1 in 
those sections. 

Again, if A = AC, then the combinations containing Ac are 
inconsistent and we eliminate them by making a figure 2 in 
those sections. 

The Reasoning Frame now shows that the definition of C is : 
C = AB I aB | ab, 
which the old logic translates, some C is B, i. e., Some S is P. 

836. Dr. Keynes gives the following examples on p. 366, of 
Formal Logic: 

All M's are P's, 
At least n S's are M's, 
Therefore, 

At least n S's are P's, 
Let B = M 
C = P 
A === at least n S's, 



§ 837.] 



AN EXAMPLE. 



463 



The propositions can be stated thus: 
B = BC, 
A = AB 

Make an ABC diagram : 



AB 


Ab 


aB 


ab 






2 






C 


1 


2 


1 




c 



Fig. 329. 

Now, if B = BC, then the combinations containing Be are 
inconsistent and we eliminate them by making a figure 1 in 
those sections. 

Again, if A = AB, then the combinations containing Ab are 
inconsistent and we eliminate them by making a figure 2 in 
those sections. 

The Reasoning Frame now shows that A = ABC, which 
we can translate: 

At least n S's are P's. 
837. "All P's are M's, 

Less than n S's are M's 
Therefore, 

Less than n S's are P's." . 
Let C = P 
B = M 

A = less than n S's, 
The premises may be stated thus: 
C = CB 
A x= AB. 



464 NUMERICAL REASONING. 

Make an ABO diagram: 



[ Chap. 29. 



AB 


Ab 


aB 


ab 






12 




1 


C 




2 






c 



Fig. 330. 

Now if = CB, then the combinations containing Cb are 
inconsistent and we eliminate them by making a figure 1 in 
those sections. 

Again, if A = AB, then the combinations containing Ab are 
inconsistent and we eliminate them by making a figure 2 in 
those sections. 

The Keasoning Frame now shows that the definition of A is : 
A == ABC | ABc, 
which the old logic would translate, 

Some less than n S's are P's. 
838. Dr. Keynes gives several other examples of numerical 
syllogisms, but they all involve more than three terms. 
Let us state the second one for an example: 
Less than n M's are P's, 
All S's are M's, 
Therefore, 

Less than n S's are P's. 
Let B = less than n M's 
C = P 
A = S 
D = M 

E = less than n S's. 
The premises can be stated thus: 
B = BO 
A = AD 
Therefore, 

E = EC. 



§ 838.] AN EXAMPLE. 

Make an ABODE diagram: 



465 



AB 


Ab 


aB 


ab 












CDE 










CDe 


2 


2 






CdE 


2 


2 






Cde 


i 




1 




cDE 


1 




1 




cDe 


2 

1 


2 


1 




cdE 


2 

1 


2 


1 




cde 



Fig. 331. 
Now, if B = BO, then the combinations containing Be are 
inconsistent and we eliminate them by making a figure 1 in 
those sections. 

Again, if A = AD, then the combinations containing Ad are 
inconsistent and we eliminate them by making a figure 2 in 
those sections. 

The Seasoning Frame now shows that the definition of E is : 
E = EC | Ec, 
which the old logic would translate, 

Some less than n S's are P's. 



30 



CHAPTER XXX. 



COMPLEX PROPOSITIONS. 



839. A complex proposition has a complex term in its sub 
ject or in its predicate or in both. 

The following are examples: 

A = B | C = D 
A | B = C | D 
Some AB == All CD 
All AB = Some CD 
All AB = All CD. 

840. Dr. Keynes says, on page 392, Formal Logic (the letter- 
ing is mine) : "Thus, taking A and C as symbols representing 
propositions, and a and c as their contradictories, the hypo- 
thetical proposition, If A then C, expresses an alternative 
between a and C and is therefore equivalent to the alternative 
proposition a or C. 

The propositions may be stated thus : 

(1) AB == ABCD 

(2) A = b | C = D. 
Make an ABCD diagram: 



AB 


Ab 


aB 


ab 


CD 

Cd 




2 




12 








12 








cD 


1 2 








cd 



Fig. 332. 
Now, if AB = ABCD, then the combinations containing 
ABCd, ABc, are inconsistent and we eliminate them by making 
a figure 1 in those sections. 



§841.] 



CONTRADICTORIES. 



467 



Again, if A = b, except where C = D, then the combina- 
tions containing AbCD, ABCd, ABc, are inconsistent and we 
eliminate them by making a figure 2 in those sections. 

I think that the proposition, A = b | C = I), can be read 
and worked backwards, but on the supposition that such is not 
the case, the result shows that the propositions are not equiva- 
lent. But (1) is an inference from (2). 

841. Let us take this example: t 

(1) Some B and C = all A, or all B is either C or both D 
and E 

(2) No A is both B and C and no B is either C or DE. 
Are these propositions contradictories? 

The propositions can be stated thus: 

(1) ABC = A, | B = C | DE 

(2) No A = BC and no B = C | DE. 
Make an ABODE diagram: 



AB 


Ab 


aB 


ab 


CDE 


2 




1 




213 




3 
3 


CDe 


213 






CUE 


213 




3 




Cde 


3 





3 

1 




cDE 


1 




cDe 


1 




1 




cdE 


1 




1 


cde 



Fig. 333. 

Now, if ABC = A, | B = C | DE, then the combinations 
containing ABCDe, ABCd, ABcDe, AB t aBcDe, aBCDE, 
aBcd, are inconsistent and we eliminate ' aem by making a 
figure 1 in those sections. 



468 COMPLEX PROPOSITIONS. [ Chap. 30. 



Again, if No A ±= BC, then the combinations containing 
ABC are inconsistent, and we eliminate them by making a 
figure 2 in those sections. 

Again, if No B = C | DE, then the combinations containing 
ABCDe, ABCd, ABcDE, aBCDe, aBCdE, aBCde, aBcDE, are 
inconsistent and we eliminate them by making a figure 3 in 
those sections. 

The Reasoning Frame now shows that the given propositions 
are contradictories, because all the B's are eliminated. 

It is very easy to get the contradictory of a complex propo- 
sition by framing an alternative proposition which has "every 
combination" for its subject and the different eliminated com- 
binations stated in the alternative for its predicate, or, by 
framing an alternative proposition with "no combination" for 
its subject and the different uneliminated combinations stated 
as alternants, for its predicate. 

842. Let us take this proposition: 

All AB = AC | DE, 
and ascertain whether it is equivalent to, 
All AB = C | DE. 

Now, if AB = AC | DE, then the combination ABCDE is 
inconsistent because it implies AB = AC and DE, and we 
therefore eliminate it by making a figure 1 in that section. 

Again, if AB === AC | DE, then the following combinations 
are inconsistent: 

ABcDe, 
ABcdE, 
ABcde, 
and we therefore eliminate them by making a figure 1 in those 
sections. 



§ 843.] EQUIVALENTS. 

Make an ABCDE diagram: 



469 



AB 


Ab 


aB 


ab 




12 









CDE 








CDe 










CdE 











Cde 








cDE 


12 








cDe 


12 








cdE 


12 








cde 



Fig. 334. 

Now, if AB = C | DE, then the combination ABCDE is 
inconsistent because it implies that AB = C and DE, and we 
therefore eliminate it by making a figure 2 in that section. 

Again, if AB = C | DE, then the following combinations 
are inconsistent: 

ABcDe, 
ABcdE, 
ABcde, 
and we therefore eliminate them by making a figure 2 in those 
sections. 

The appearance of the Keasoning Frame shows that the 
two propositions are equivalent. This example is taken from 
Dr. Keynes' "Formal Logic," p. 396. 

843. Let us take the three following propositions and ascer- 
tain whether they are equivalent to each other: 

(1) All AB = a | 

(2) All B = a | C 

(3) AB = C. 



470 COMPLEX PROPOSITIONS. 

Make an ABC diagram: 



[ Chap. 30. 



AB 


Ab 


aB 


ab 













C 


12 




c 



Fig. 335. 

Now, if AB — a | C, then the combination ABc is incon- 
sistent and we eliminate it by making a figure 1 in that section. 

Again, if AB = C, then the combination ABc is inconsistent 
and we eliminate it by making a figure 2 in that section. 

The result shows that (1) and (3) are equivalent. 

Make another ABC diagram: 



AB 


Ab 


aB 


ab 








2 




C 


1 








c 



Now, if B 



Fig. 336. 
C, then the combination ABc, is inconsist- 



ent because it contains neither a nor C, and we therefore elim- 
inate it by making a figure 1 in that section. 

Again, if B = a | C, then the combination aBC is incon- 
sistent because it means B = a, and C, and we therefore elim- 
inate it by making a figure 2 in that section. 

The result shows that (2) is not equivalent to either (1) or 
(3). (1) and (3) can be inferred from (2). 

844. Let us take this example: 

Are the following propositions equivalent? 

(1) A | B = e 

(2) A = AC and B = BC. 



§ 845.] EQUIVALENTS. 

Make an ABC diagram: 



471 



AB 


Ab 


aB 


ab 












C 


23 


12 


13 




c 



Fig. 337. 

Now, if A | B = C, then the combinations Abe, aBc, are 
inconsistent and we eliminate them by making a figure 1 in 
those sections. 

Again, if A = AC, then the combinations containing Ac are 
inconsistent and we eliminate them by making a figure 2 in 
those sections. 

Again, if B = BC, then the combinations containing Be are 
inconsistent and we eliminate them by making a figure 3 in 
those sections. 

The Reasoning Frame now shows that the given propositions 
are not equivalents. 

845. Let us take this example: 

Are the following propositions equivalents? 

(1) A = B | C 

(2) A = B | A = C. 
Make an ABC diagram: 



AB 


Ab 


aB 


ab 




213 








C 




213 






c 



Fig. 338 



472 



COMPLEX PROPOSITIONS. 



[ Chap. 30. 



Now, if A = B | C, then the combinations ABO, Abe, are 
inconsistent and we elimiDate them by making a figure 1 in 
those sections. 

Again, if A = B, except where A = C, then the combina- 
tions ABC, Abe, are inconsistent and we eliminate them by 
making a figure 2 in those sections. 

Again if A = C, except where A == B,then the combinations 
ABC, Abe, are inconsistent and we eliminate them by making 
a figure 3 in those sections. 

The Reasoning Frame now shows that the given propositions 
are equivalent. 

(Keynes, "Formal Logic," p. 398.) 

846. Let us take this example : 

Are the following propositions equivalent? 

(1) No A = C | No B = C 

(2) No AB = C. 
Make an ABC diagram, 



AB 


Ab 


aB 


ab 




3 


1 


2 




C 










c 



Fig. 339. 

Now, if No A = C, except where No B = C, then the com- 
bination AbC is inconsistent and we eliminate it by making a 
figure 1 in that section. 

Again, if No B = C, except where No A = C, then the 
combination aBC is inconsistent and we eliminate it by mak- 
ing a figure 2 in that section. 

Again, if No AB = C, then the combination ABC, is incon- 
sistent and we eliminate it by making a figure 3 in that section. 

The Reasoning Frame now shows that the given propositions 
.are not equivalents. 

847. Let us take this example: 



§848.] 



INFERENCE. 



473 



Given the proposition A | B = CD, can we infer that all 
A = C? 

The propositions can be stated thus: 

(1) Ab | aB == CD 

(2) A = AC. 
Make an A BCD diagram: 



AB 


Ab 


aB 


ab 












CD 




1 


1 




Cd 




1 


1 




cD 




1 


1 




cd 



Fig. 340. 

Now, if A | B = CD, then the combinations containing 
AbCd, Abe, aBCd, aBc, are inconsistent and we eliminate them 
by making a figure 1 in those sections. 

From the uneliminated combinations we can get this defini- 
tion of A : 

A = AB | AbC. 
This shows that we cannot infer that A = C. 
848. Let us take this example : 
From the proposition, 

(1) CD = ABCD 
can we infer the proposition, 

(2) C = AC? 



474 COMPLEX PROPOSITIONS. 

Make an ABCD diagram: 



[ Chap. 30. 



AB 


Ab 


aB 


ab 






1 


12 


12 


CD 






2 


2 


Cd 










cD 










cd 



Fig. 341, 

Now, if CD = ABCD, then the combinations containing 
AbCD, aCD, are inconsistent and we eliminate them by making 
a figure 1 in those sections. 

Again, if C — AC, then the combinations containing aC are 
inconsistent and we eliminate them by making a figure 2 in 
those sections. 

The Reasoning Frame now shows that: 

(1) The given propositions are not equivalents. 

(2) They are consistent. 

(3) Neither can be inferred from the other. 
849. Let us take the following example: 

Is the proposition, 

No A = C, equivalent to the following proposition? 
No Ab | aB =Cd | cD. 
Make an ABCD diagram: 



AB 


A>> 


aB 


ab 




1 


i 






CD 


1 


12 


9 
2 




Cd 




2 




cl) 










cd 



Fig. 342. 
Now, if No A = C, then the combinations containing AC are 



§ 850.J 



EQUIVALENTS. 



475 



inconsistent and we eliminate them by making a figure 1 in 
those sections. 

Again, if No Ab | aB = Cd | cD, then the combinations 
containing AbCd, AbcD, aBCd, aBcD, are inconsistent and we 
eliminate them by making a figure 2 in those sections. 

The result now shows that the given propositions are not 
equivalent. 

850. Let us take this example : 

Given the proposition, 

(1) All A = BC | DE, 
is it equivalent to the proposition, 

(2) No A = bd | be | cd | ce? 
Make an ABODE diagram: 



AB 


Ab 


aB 


ab 




1 








CDE 




21 






CDe 




21 






CdE 




1 






Cde 










cDE 


21 


21 






cDe 


21 


1 






cdE 


1 


1 






cde 



Fig. 343. 

Now, if A = BC | DE, then the combinations containing 
ABCDE, ABcDe, ABcd, AbCDe, AbCd, AbcDe, Abed, are 
inconsistent and we eliminate them by making a figure 1 in 
those sections. 

Again, if No A = bd be cd ce, then the combi- 
nations containing AbCDe, AbCdE, AbcDe, ABcDe, ABcdE are 
inconsistent and we eliminate them by making a figure 2 in 
those sections. 



476 



C0MPL1X PROPOSITIONS. 



[ Chap. 30. 



The result proves that the given propositions are not equiva- 
lent, but (2) can be inferred from (1). 

851. Let us take this example: 
Given the following proposition, 

(1) AB is not either C or D, 
can we infer that, 

(2) A = Ac? 
Make an ABCD diagram: 



AB 


Ab 


aB 


ab 




2 


2 




CD 


12 


2 






Cd 


1 






cD 










cd 



Fig. 344. 

Now, if AB = not either C | D, then the combinations con- 
taining ABCd, ABcD, are inconsistent and we eliminate them 
by making a figure 1 in those sections. I assume that or is 
exclusive. 

Again, if A = Ac, then the combinations containing AC are 
inconsistent and we eliminate them by making a figure 2 in 
those sections. 

The Reasoning Frame now shows that the given propositions 
are inconsistent, because the combination ABCD is eliminated, 
and this combination is necessary to the expression of the 
given proposition, AB is not either C or D. 

852. Dr. Keynes gives on p. 400, "Formal Logic," the follow- 
ing examples : 

(1) All A is CD, therefore, All AB is C, 

(2) No A is C, therefore, No AB is CD. 

(3) Given All A is C, then, All AB is C by rule (1) above; 
and from this we obtain All AB is BC by rule (2) of 
section 338. 



§§ 853, 854.] 



INFERENCE. 



477 



Make an ABCD diagram : 



AB 


Ab 


aB 


ab 










CD 


1 


1 






Cd 


1 


1 






cD 


1 


1 






cd 



Fig. 345. 
. Now, if All A = CD, then the following combinations are 
inconsistent, ABCd, ABcD, ABcd, AbCd, AbcD, Abed, and 
we eliminate them by making a figure 1 in those sections. 

The Reasoning Frame now shows that All AB — C, there- 
fore, (1) is correct. 

853. Make another ABCD diagram : 



AB 


Ab 


aB 


ab 




1 


1 







CD 


1 


1 




Cd 










cD 










cd 



Fig. 346. 

Now, if No A = C, then the following combinations are 
inconsistent, ABCD, ABCd, AbCD, AbCd, and we eliminate 
them by making a figure 1 in those sections. 

The Reasoning Frame now shows that No AB = CD, there- 
fore, (2) is correct. 

854. The proposition, Given All A is C, may be stated thus : 
A = AC. 

Now, if A = AC, then the combinations containing Ac are 



478 



COMPLEX PROPO IFIONS. 



[Chap. 30. 



inconsistent and we eliminate them by making a figure 1 in 
those sections. 

We can now read in the Eeasoning Frame, All AB — C, 
and All AB — BC, therefore, (3) is correct. 

Make an ABC diagram and eliminate as above directed. 



AB 


Ab 


aB 


ab 












C 


1 


1 






c 



Fig. 347. 

855. Dr. Keynes also gives the following examples on p. 
401, of Formal Logic: 

(1) Given No A is C, then No AB is C, and, therefore, by 
rule (5) of section 338, No AB is b or C. 

(2) Given No A is C, then No A is BC, and, therefore, by rule 
(6) of section 338, No A or b is BO. 

Make an ABC diagram : 



AB 


Ab 


aB 


ab 




1 


1 






C 










c 



Fig. 348. 

Now, if No A = C,then the combinations containing AC are 
inconsistent and we eliminate them by making a figure 1 in 
those sections. 

We can now read in the Reasoning Frame, 
No AB = C. 



§ 855.] 



INFERENCE. 



479 



The term b or C, means b without C or C without b. It can 
be expressed thus : 

be | BO 

Now, if No AB = be | BC, then the combination ABC is 
inconsistent and we eliminate it by making a figure 1 in that 
section. 

Make an ABC diagram: 



AB 


Ab 


aB 


ab 




1 








C 










c 



Fig. 349. 
We can now read in the Reasoning Frame the proposition, 
Given No A = C, then. No AB = b | 0. 
Make an ABC diagram: 



AB 


Ab 


aB 


ab 




1 


1 






C 










c 



Fig. 350. 

Now, if No A = C, then the combinations containing AC are 
inconsistent and we eliminate them by making a figure 1 in 
those sections. 

The term A or b means A without b or b without A, and can 
be expressed thus : 

AB J ab. 
We can now read in the Reasoning Frame, 

No A = BC, and also No AB | ab = BC. (1) and (2) 
are correct. 



4S0 COUPLIX PROPOSITIONS. [ Chap. 30. 

856. In solving problems. I use stiff pieces of cardboard on 
which are drawn the different diagrams, and two kinds of 
counters to place in the sections. One kind signifies, when 
placed in a section, that the section is eliminated. The other 
kind, when placed in a section, signifies that the section is 
saved. 

When I have a disjunctive proposition with a number of 
alternants, for example. A is B or G or D, I put a counter 
which signifies that a section is saved in every combination 
which contains A. and also contains B or C or D, or a combina- 
tion of them. 

In the remaining A sections I put a counter which signifies 
that the combination in that section is eliminated. 

Next. I proceed to eliminate the combinations containing A, 
which also contain B and C, and I put in the sections contain- 
ing those combinations, counters which signify that the com- 
binations are eliminated. 

Next, in a combination containing A. which also contains B 
and P. I take up the saving counter and put in its place an 
eliminating counter. 

Xext. in a section containing A. which also contains C and 
D. I take up the saving counter and put in its place an eliminat- 
ing counter. 

My Reasoning Frame now shows me all of the A combina- 
tions which are eliminated and all which are saved. 

The two kinds together make all of the A combinations. 

This method of getting the logical expression of a proposi- 
tion is very simple and easy. 

857. Let us take this example: 
Can we infer from the proposition, 

(1) BC == ABC, that, 

(2) AC = ABC? 



858.] EQUIVALENTS. 

Make an ABC diagram : 



481 



AB 


Ab 


aB 


ab 






2 


1 




C 










c 



Fig. 351. 

Now, if BC = ABC, then the combination aBC, is incon- 
sistent and we eliminate it by making a figure 1 in that section. 

Again, if AC = ABC, then the combination AbC, is incon- 
sistent and we eliminate it by making a figure 2 in that section. 

The Reasoning Frame now shows: 

(1) The given propositions are not equivalents, 

(2) They are consistent. 

(3) Neither can be inferred from the other. 
858. Let us take this example : 

Are the following propositions equivalents? 

(1) ABC == BC 

(2) ABC = AC. 
Make an ABC diagram: 



AB 


Ab 


aB 


ab 






2 


1 




C 










c 



Fig. 352. 

Now, if BC = ABC, then the combination aBC is incon- 
sistent and we eliminate it by making a figure 1 in that section. 

Again, if AC = ABC, then the combination AbC, is incon- 
sistent and we eliminate it by making a figure 2 in that section, 

31 



482 



COMPLEX PROPOSITIONS. 



[ Chap. 30. 



The result proves that the two propositions are not equiv- 
alent. 

859. Let us take this example : 

What propositions are consistent with the following propo- 
sition? 

BC | BD = A. 

Make an ABCD diagram: 



AB 


Ab 


aB 


ab 












CD 






1 




Cd 






1 




cD 










cd 



Fig. 358 

Now, if BO | BD == A, then the combinations aBCd, aBcD, 
are inconsistent and we eliminate them by making a figure 1 in 
those sections. 



We can now read 



1) aBO = aBCD 

2) aBc = aBcd 

3) Cd == CdA | Cdab 

4) cD == cDA j cDab 

5) No a = BCd 

6) No a == BcD 

7) No B = aCd 

8) No B = acD 

9) No C = aBd 

10) No c = aBD 

11) NoD = aBc 

12) No d = aBC 

13) aB = CD | cd 
U) | D = A | ab. 



§§ 860, 861.] 



INFERENCE. 



483 



Many other readings could be given but they would be 
simply trifling variations of those already given. 
860. Let us take the following example: 
From the proposition, 

(1) A = ABC 
can we infer the proposition, 

(2) a = b J c? 
Make an ABC diagram: 



AB 


Ab 


aB 


ab 






1 


2 




C 


1 


1 




2 


c 



Fig. 354. 

Now, if A = ABC, then the combinations containing ABc, 
Ab, are inconsistent, and we eliminate them by making a fig- 
ure 1 in those sections. 

Again, if a = b | c, then the combinations containing aBC, 
abc, are inconsistent, and we eliminate them by making a fig- 
ure 2 in those sections. 

The Reasoning Frame now shows : 

(1) The given propositions are not equivalents 

(2) They are consistent 

(3) Neither can be inferred from the other 
861. Let us take this example: 

Are the following propositions equivalents? 

(1) No A = B | 

(2) A = Abc 



484 COMPLEX PROPOSITIONS. 

Make an ABC diagram: 



[ Chap. 80. 



AB 


Ab 


aB 


ab 




2 


12 






C 


21 








c 



Fig. 355. 

Now, if No A — B | C, then the combinations ABc, AbC, are 
inconsistent, and we eliminate them by making a figure 1 in 
those sections. 

Again, if A = Abe, then the combinations containing AB, 
AbC, are inconsistent, and we eliminate them by making a fig- 
ure 2 in those sections. 

The Reasoning Frame now shows that the given proposi- 
tions are not equivalents. 

862. Let us take the following example: 

From the proposition, 

(1) AB = ABC 
can we infer the proposition, 

(2) A = b | 0? 
Make an ABC diagram : 



AB Ab 


aB 


ab 






2 






C 


21 








c 



Fig. 356. 

Now, if AB = ABC, then the combination ABc is inconsist- 
ent, and we eliminate it by making a figure 1 in that section. 
Again, if A = b | C, then the combinations containing AbC, 



§§ 863, 864.] 



INFERENCE. 



485 



ABc, are inconsistent, and we eliminate them by making a fig- 
ure 2 in those sections. 

The Reasoning Frame now shows : 

(1) The given propositions are not equivalents 

(2) They are consistent 

(3) (1) can be inferred from (2) 

(4) (2) cannot be inferred from (1) 
863. Let us take the following example: 
From the proposition, 

(1) A = B | C 
can we infer the proposition, 

(2) Ab = AbC? 
Make an ABC diagram : 



AB 


Ab 


aB 


ab 




1 








C 




21 






c 



Fig. 357. 

Now, if A = B | C, then the combinations ABC, Abe, are 
inconsistent, and we eliminate them by making a figure 1 in 
those sections. 

Again, if Ab = AbC, then the combination Abe is inconsist- 
ent, and we eliminate it by making a figure 2 in that section. 

The Reasoning Frame now shows: 

(1) The given propositions are not equivalents 

(2) They are consistent 

(3) (2) can be inferred from (1) 

(4) (1) cannot be inferred from (2) 
(Keynes' "Formal Logic," p. 407). 

864. Let us take the following example: 



486 



COMPLEX PROPOSITIONS. 



[ Chap. 30. 



Are the following propositions equivalents? 

(1) AB = CD | de 

(2) cD | dE = a | b 
Make an ABCDE diagram : 



AB 


Ab 


aB 


ab 












CDE 










CDe 


12 






2 


CdE 










Cde 


12 






2 


cDE 


12 






2 


cDe 


12 






2 


cdE 










cde 



Fig. 358. 

Now, if AB = CD | de, then the combinations containing 
ABCdE, ABcD, ABcdE, are inconsistent, and we eliminate 
them by making a figure 1 in those sections. 

Again, if cD or dE = a or b, then the combinations contain- 
ing ABCdE, ABcD, ABcdE, abCdE, abcD, abcdE, are incon- 
sistent, and we eliminate them by making a figure 2 in. those 
sections. 

The Reasoning Frame now shows that the given Droposi- 
tions are not equivalents. 

iWe can infer (1) from (2), but not conversely. 
j 865. Let us take the following example: 

l Kre the following propositions equivalents? 

(1) No AB = CD | EF 

(2) No A = BCD | BEF 



866.] EQUIVALENTS. 

Make an ABCDEF diagram: 



487 



ABC 


ABc 


AbC 


Abe 


aBC 


aBc 


abC 


abc 






1 














DEF 


1 
















DEf 


1 
















DeF 


1 
















Def 


1 


1 














dEF 


















dEf 


















deF 


















def 



Fig. 359. 

Now, if No AB = CD | EF, then the combinations contain- 
ing ABCDEf, ABCDe, ABCdEF, ABcEF, are inconsistent, and 
we eliminate them by making a figure 1 in those sections. 
The Reasoning Frame now shows that, 

No A = BCD | BEF, 
and the given propositions are equivalents. 

(Keynes' "Formal Logic," p. 409.) 
866. Dr. Keynes gives the following example on p. 409 of 
"Formal Logic": 

(1) No AB is CD or EF, therefore, 

No A is BCD or BEF 

No C is ABD or ABEF 

No BD is AC or AEF 

The proposition No AB is CD or EF, can be stated thus: 

No AB = CD I EF 



488 COMPLEX PROPOSITIONS. 

Make an ABCDEF diagram : 



[ Chap. 30. 



ABC 


ABc 


AbC 


Abe 


aBC 


aBc 


abC 


abc 






1 














DEF 


1 
















DEf 


1 
















DeF 


1 
















Def 


1 


1 














dEF 


















dEf 


















deF 


















iief 



Fig. 360. 

Now, if No A B = CD | EF, then the following combinations 
ABCDEf, ABCDeF, ABCDef, ABCdEF, ABcDEF, ABcdEF, 
are inconsistent, and we eliminate them by making a figure 1 in 
those sections. 

The Reasoning Frame now shows the logical expression of 
No AB = CD | EF. 

We can also read in it, No A = BOD | BEF, and No C = 
ABD | ABEF, and No BD = AC | AEF. 

867. Let us take the following example: 

Can we infer the proposition, 
All F == ABDE 
from the propositions, 

(1) F = AB | bee, 

(2) F = aBC | DE? 



§ 868.] INFERENCE. 

Make an ABCDEF diagram: 



489 



ABC 


ABc AbC 


Abe 


aBC 


aBc 


abC 


abc 








12 


21 


1 


12 


21 


21 


DEF 


















DEf 


2 


2 


12 


2 


1 


12 


21 


2 


DeF 


















Def 


2 


2 


12 


12 


1 


12 


21 


21 


dEF 


















dEf 


2 


2 


12 


2 


1 


12 


21 


2 


deF 


















def 



Fig. 361. 

Now, if F = AB | bee, then the combinations containing 
AbCF, AbcEF, aBF, abCF, abcEF, are inconsistent, and we 
eliminate them by making a figure 1 in those sections. 

Again, if F = aBC | DE, then the combinations containing 
abF, aBcF, AbF, ABCDeF, ABCdF, ABcDeF, ABcdF, are 
inconsistent, and we eliminate them by making a figure 2 in 
those sections. 

The Reasoning Frame now shows that, 
All F = ABDE 

868. Let us take this example: 

Can the proposition, 

No A = be | Cd 
be inferred from the propositions, 

(1) No A = be 

(2) No A = Cd? 



490 COMPLEX PROPOSITIONS. 

Make an ABCD diagram: 



[Chap. 30. 



AB 


Ab 


aB 


ab 




2 








CD 


2 






Cd 


1 






cD 




1 






cd 



Fig. 362. 

Now, if No A = be, then the combinations containing Abe 
are inconsistent and we eliminate them by making a figure 1 in 
those sections. 

Again, if No A = Cd, then the combinations containing ACd 
are inconsistent and we eliminate them by making a figure 2 in 
those sections. 

The Reasoning Frame now shows that No A = be | Cd, i. e., 
bcD | BCd. 

869. Let us take the following example : 

Can we infer the proposition, 

D = A | Be 
from the propositions, 

(1) D = A | B 

(2) No D = aC? 
Make an ABCD diagram: 



AB 


Ab 


aB 


ab 




1 




3 
2 


3 
21 


CD 










Cd 


13 






3 I 


cD 










cd 



Fig. 363. 



§§ 870-877.] INFERENCE. 491 

Now, if D = A | B, then the combinations containing ABD, 
abD, are inconsistent, and we eliminate them by making a fig- 
ure 1 in those sections. 

Again, if No D = aC, then the combinations containing aCD 
are inconsistent, and we eliminate them by making a figure 2 in 
those sections. 

Again, if D = A | Be, then the combinations containing 
ABcD, aCD, abcD, are inconsistent, and we eliminate them by 
making a figure 3 in those sections. 

Now, as the given inference can be read in the Reasoning 
Frame, and it does not eliminate any combinations which the 
given propositions did not eliminate, this proves that the given 
inference can be inferred from the given propositions. 

870. Propositions can be divided into two groups, viz. : 

(1) Those which are consistent, 

(2) Those which are inconsistent. 

871. Consistent propositions may be again divided into, 

(1) Those which are consistent only, 

(2) Those which stand in the relation of inferend 

and inference, 

(3) Those which are equivalent. 

872. Inconsistent propositions may be divided into three 
groups, viz.: 

(1) Those which are inconsistent only, 

(2) Those which are contradictories, 

(3) Those which are perfect contradictories. 

873. An Inferend proposition is a proposition from which 
another can be inferred, which is called the Inference. 

874. To ascertain whether two propositions are consistent, 
we get the visible expression of both in the proper Reasoning 
Frame. 

875. If both can now be read in the Reasoning Frame, then 
they are consistent. 

876. If both eliminate exactly the same combinations, then 
they are equivalents. 

877. If one of them eliminates more combinations than the 
other, and if the one which eliminates the fewer combinations 



492 COMPLEX PROPOSITIONS. [ Chap. £0. 

does not eliminate any combinations excepting those which 
the other eliminated, and both can be read in the Seasoning 
Frame, then these two propositions stand to each other in the 
relation of inferend and inference. 

The one which eliminated the greater number of combina- 
tions is the Inferend. 

The one which eliminated the lesser number is the Inference. 

878. If two given propositions eliminate entirely different 
combinations, and can both be read in the Reasoning Frame, 
then they are merely consistent, but one is not an inference 
from the other. 

879. To ascertain whether two propositions are inconsis- 
tent, we get the visible expression of both in the Reasoning 
Frame. 

880. If both cannot now be read in the Reasoning Frame, 
then they are inconsistent. 

881. If a letter-term has been eliminated, then they are con- 
tradictories. 

882. If one eliminates the combinations which the other 
saved and saves the combinations which the other eliminated, 
then they are perfect contradictories. 

883. Let us take the following example: 
Can we infer the proposition, 

NoA = BD | BE | CD | CE 
from the propositions, 

(1) No A = B | C 

(2) No A = D | E? 



884.] INFERENCE. 

Make an ABODE diagram: 



493 



AB 


Ab 


aB 


ab 






1 


1 




CDE 


2 


3 
21 


] 




CDe 


2 


3 
21 


1 




CdE 




1 


1 




Cde 










cDE 


3 

2 


2 






cDe 


3 
2 


2 






cdE 










cde 



Fig. 364. 

Now, if No A = B | 0, then the combinations containing 
AbC, aBC, are inconsistent, and we eliminate them by making 
a figure 1 in those sections. 

Again, if No A = D | E, then the combinations containing 
ACDe, ACdE, AcDe, AcdE, are inconsistent, and we eliminate 
them by making a figure 2 in those sections. 

Again, if No A = BD | BE | CD | CE, then the combinations 
containing ABcDe, ABcdE, AbCDe, AbCdE, are inconsistent, 
and we eliminate them by making a figure 3 in those sections. 

The Reasoning Frame now shows that we can infer the 
given inference from the given propositions, because the given 
inference can be read, and it eliminates combinations' only 
which have already been eliminated, and does not eliminate any 
combinations which the given propositions did not eliminate. 

884. Let us take the following example. Can we infer the 
proposition, 

A = B | C 
from the proposition, 

(1) A = B I A = C? 



494 COMPLEX PROPOSITIONS, 

Make an ABC diagram: 



[ Chap. 30. 



AB 


Ab 


aB 


ab 




12 








C 




12 






c 



Fig. 365. 

Now, if A = B | A = C, then the combinations ABC, Abe, 
are inconsistent, and we eliminate them by making a figure 1 
in those sections. 

Again, if A = B | C, then the combinations ABC, Abe, are 
inconsistent, and we eliminate them by making a figure 2 in 
these sections. 

The Reasoning Frame now shows: 

(1) That the given inference can be inferred from the 
given inferend, and conversely, that the given inferend 
can be inferred from the given inference. 

(2) That the given inferend and the given inference are 
equivalents. 

885. Let us take the following example: Can we infer the 
proposition 

No A — BC | be 
from the proposition 

(1) A — b ] A == c? 
Make an ABC diagram : 



AB 


Ab 


aB 


ab 




23 









C 




13 


c 



Fig. 366. 



§ 886.] 



INFERENCE. 



495 



Now, if A = b, except where A = c, then the combination 
Abe is inconsistent, and we eliminate it by making a figure 1 in 
that section. 

Again, if A = c, except where A = b, then the combination 
ABC is inconsistent, and we eliminate it by making a figure 2 
in that section. 

Again, if No A = BC | be, then the combinations ABC, Abe, 
are inconsistent, and we eliminate them by making a figure 3 
in those sections. 

The Keasoning Frame now shows: 

(1) The given propositions are equivalent. 

(2) Each can be inferred from the other. 

(3) They are consistent. 

886. Let us take the following example: Can we infer the 
proposition, 

Something (i. e. some combinations) = AB | CD 

from the proposition, 

(1) A = B | = D? 
Make an ABCD diagram : 



AB 


Ab 


aB 


ab 




12 








CD 




12 


3 


2 


Cd 




1 






cD 




1 






cd 



Fig. 367. 

Now, if A = B, except where C = D, then the combinations 
containing ABCD, AbCd, Abe, are inconsistent, and we elimi- 
nate them by making a figure 1 in those sections. 

Again, if C = D, except where A = B, then the combina- 
tions containing ABCD, AbCd, aCd, are inconsistent, and we 
eliminate them by making a figure 2 in those sections. 



4C6 



COMPLEX PROPOSITIONS. 



[Chap. 30. 



The Reasoning Frame now shows that we can read in it, 

(1) Something = AB | CD. 

(2) Something = ac. 

887. The following example is taken from Keynes' "Formal 
Logic," p. 425 : "Given, 

1st. That wherever the properties A and B are combined, 
either the property C, or the property D, is present also, but 
they are not jointly present. 

2d. That wherever the properties B and C are combined, 
the properties A and D are either both present or both absent. 

3d. That wherever the properties A and B are both absent, 
the properties C and D are both absent also; and vice versa, 
where the properties C and D are both absent, A and B are 
both absent also. 

Find what can be inferred from the presence of A with 
regard to the presence or absence of B, C, and D." 

The premises may be stated thus : 

(1) AB = Cd | cD 

(2) BC = AD | ad 

(3) ab = cd 

(4) cd = ab 
Make an ABCD diagram : 



AB 


Ab 


aB 


ab 




1 




2 


3 


CD 


3 






3 


Cd 








3 


cD 


14 


4 


4 




cd 



Fig. 368. 

Now, if AB = Cd | cD, then the combinations ABCD, ABcd, 
are inconsistent, and we eliminate them by making a figure 1 
in those sections. 

Again, if BC === AD | ad, then the combinations ABCd, 



§ 888.] INFERENCE. 497 

aBCD, are inconsistent, and we eliminate them by making a 
figure 2 in those sections. 

Again, if ab = cd, then the combinations abCD, abCd, abcD, 
are inconsistent, and we eliminate them by making a figure 3 
in those sections. 

Again, if cd = ab, then the combinations ABcd, Abed, aBcd, 
are inconsistent, and we eliminate them by making a figure 4 
in those sections. 

The Reasoning Frame now gives us the following combina- 
tions containing A, B, C, D: 

(1) ABcD 
which can be translated, 

When B is present with A, C is absent and D is present. 

(2) AbCD 
Which can be translated, 

When C and D are present with A, B is absent; or 

(3) AbCd 
which can be translated, 

When B and D are both absent from A, C is present; or 

(4) AbcD 
which can be translated, 

When B and C are both absent from A, D is present. 
This information can be sumed up thus : 

Where A is, B is absent and C is present, or C is absent 
and D is present. 
888. Let us take the following example: 

Can we eliminate B, together with b, from the proposi- 
tions, 

All A = BC | bD 

Whatever is B I D = a I BCD? 



33 



498 COMPLEX PROPOSITIONS. 

Make an ABCD diagram: 



[ Chap. 30. 



AB 


Ab 


aB 


ab 






2 






( D 


2 


1 






Cd 


1 


2 






cD 


21 


1 






cd 



Fig. 369. 

Now, if A = BC | bD, then the combinations ABcD, ABcd, 
AbCd, Abed, are inconsistent, and we eliminate them by 
making a figure 1 in those sections. 

Again, if B | D = a | BCD, then the combinations ABCd, 
ABcd, AbCD, AbcD, are inconsistent, because they imply that 
B | D — A, and we therefore elimisate them by making a 
figure 2 in those sections. 

The Reasoning Frame now shows us that B is not eliminated 
from the given proposition, but that b is eliminated. But if 
B were eliminated, then all the A's would be eliminated, and 
this would prove the contradictoriness of the premises. 

889. The following example is adapted from an example 
given by Dr. Keynes, on p. 429 : 

Nothing = ac ] bO, 
therefore, 

Nothing = ab. 

Make an ABC diagram : 



AB 


Ab 


aB 


ab 






1 




1 


U 






1 


1 


c 



Fig. 370. 



§ 889.1 INFERENCE. 499 

Now, if Nothing (i. e. no combinations) =5 ac | bC, then the 
combinations containing ac, bC, are inconsistent, and we elimi- 
nate them by making a figure 1 in those sections. 

The Reasoning Frame now shows that we can read in it, 
Nothing = ab 
Therefore, the given conclusion is correct. 

I understand that in this connection nothing means no com- 
binations. 



CHAPTER XXXI. 



EXAMPLES. 



890. Let us suppose the following state of facts: 

One Sunday morning five thieves went to the town of L — , 
and stayed there one week. They committed a burglary each 
night. Three, and three only, went out together. We will 
call them A, B, C, D, E. Commencing Sunday night, they 
went out in the inverse order of their names. B always went 
out with C, D, or E. On Saturday night, just before leaving 
town, the three who went out that night committed a murder. 

From these facts, tell which three committed the murder. 

Make an ABCDE diagram: 



AB 


Ab 


aB 


ab 




1 


1 


1 




CDE 


1 






1 


CDe 


1 






1 


CdE 


2 


1 


1 


1 


Cde 


1 






1 


cDE 


2 


1 


1 


1 


cDe 


2 


1 


1 


1 


cdE 


1 


1 


1 


1 


cde 



Fig. 371. 

Now, if three, and three only, went out together, then all the 
combinations which contain either more or less than three 
capital letters will be inconsistent, and we eliminate them by 
making a figure 1 in those sections. 



§ 891 .] THE YACHT PROBLEM. 501 

An uneliminated combination containing three capital let- 
ters may contain the names of the three who went out 
together. 

Again, if B went out only with C, D, or E, then all the 
remaining uneliminated combinations containing A and B are 
inconsistent, and we eliminate them by making a figure 2 in 
those sections. 

The uneliminated combinations, taking them in inverse 
order, are ODE, BDE, BCE, BCD, ADE, ACE, ACD. There- 
fore the three who committed the murder were A, C, and D. 

891. Let us take the following supposed state of facts: 

Six rich men, whom we will call A, B, C, D, E, F, chartered a 
steam yacht for not to exceed seven trips, for two thousand 
dollars, and they agreed to divide the expenses according to 
the number of rides each one should take. But they kept no 
record of their trips, and some time afterward, when they 
came to have a final settlement, they hopelessly disagreed as 
to the number of trips which had been taken, and as to who 
went on each trip. A lawsuit resulted. On the trial the testi- 
mony was very conflicting, and in addition to the facts above 
pven, the following facts only were proven : 

(1) Four, and four only, went on each trip. 

(2) Taken together in couples, A and B, A and C, B and C* 
iever went on the same trip with D and E or D and F or E and 

f , and C and D never went with E and F. 

From these facts, tell what each man's share of the expenses 
wa* 



502 EXAMPLES. 

Make an ABCDEF diagram : 



[Chap. 31. 



ABC 


ABc 


AbC 


Abe 


aBC 


aBc 


abC 


abc 




1 


1 


1 




1 




5 


1 


DEF 


1 


2 


3 




4 






1 


DEf 


1 


2 


8 




4 






1 


DeF 




1 


1 




1 






1 


Def 


1 


2 


3 




4 






1 


dEF 




1 


1 




1 






1 


dEf 




1 


1 




1 






1 


deF 


1 


1 


1 




1 






1 


def 



Fig. 372. 

Let capital letters represent the men on a trip. Now if four, 
and only four, went on a trip, then all the combinations con- 
taining either more or less than four capital letters are incon- 
sistent, and we eliminate them by making a figure 1 in those 
sections. 

Again, as A and B never went with D and E, or D and F, or 
E and F, we eliminate the remaining ABDE, ABDF, ABEF 
combinations by making a figure 2 in those sections. 

Again, as A and C never went with D and E, or D and F, or 
E and F, we eliminate the combinations remaining which con- 
tain ACDE, ACDF, ACEF, by making a figure 3 in those 
sections. 

Again, as B and C never went with D and E, or D and F, or 
E and F, we eliminate the remaining combinations which con- 
tain BCDE, BCDF, BCEF, by making a figure 4 in those sec- 
tions. 

Again, as C and D never went with E and F, we eliminate 
the remaining CDEF combination by making a figure 5 in 
that section. 



§ 892.] 



THE YOUNG LADIES PROBLEM. 



503 



The uneliminated combinations will give us the number of 
the trips taken, and the names of the men who went on each 
trip. 

They are ABCD, ABCE, ABCF, ADEF, BDEF. 

An examination of these combinations shows that A went 
four times, B four times, C three times, D three times, E three 
times, and F three times. Therefore, A's share was four hun- 
dred dollars, B's four hundred dollars, and each of the others 
three hundred dollars. 

892. Suppose the following facts to be granted: 

Four young ladies, whom we will call A, B, C,and D,were in 
the habit of taking long walks for exercise. It was noticed 
that C and D never went out with A or B, and that they never 
stayed in with A or B; that C without D never went out, or 
stayed in with A and B ; and that D without C never went out, 
or stayed in with A and B. One or more of them went out 
every day of the week, but the same ones never went together 
twice in the same week. 

The problem is to tell which ones took walks together during 
the week. 

Let capital letters represent those who went out walking. 

Let small letters represent those who stayed in the house. 

Make an ABCD diagram : 



AB 


Ab 


aB 


ab 






1 


1 




CD 


2 






5 


Cd 


3 






5 


cD 




4 


4 




cd 



Fig. 373. 

Now, if C and D never went out with A or B, then the com- 
binations AbCD, aBCD, are inconsistent, and we eliminate 
them by making a figure 1 in those sections. 

NoWj if C without D, never went out with A and B, then the 



504 



EXAMPLES. 



[ Chap. 31, 



combination ABCd is inconsistent, and we eliminate it by mak- 
ing a figure 2 in that section. 

Again, if D without C never went out with A and B, then the 
combinations ABcD is inconsistent, and we eliminate it by 
making a figure 3 in that section. 

Again, if C and D never stayed in with A or B, then the com- 
binations Abed, aBcd, are inconsistent, and we eliminate them 
by making a figure 4 in those sections. 

Again, if A and B never stayed in with C or D, then the com- 
binations abCd, abcD, are inconsistent, and we eliminate them 
by making a figure 5 in those sections. 

The uneliminated combinations show who went out 
together. 

Omitting the small letters, they are: A, B, C, and D, A and 
B, A and C, A and D, B and C, B and D, C and D. 

893. Let us suppose the following facts: 

A party of gentlemen whom we will call A, B, 0, and D, 
went a-fishing on four different occasions. and D never 
fished with A and B,or without A or B,C and D fished together 
or did not fish at all. The same ones never fished together 
more than once. 

The problem is to tell who fished on the four different occa- 
sions. 

Let capital letters represent the men when they went out 
a-fishing. 

Let small letters represent the men when they stayed in. 

Make an ABCD diagram: 



AB 


Ab 


aB 


ab 




1 






1 


CD 


3 


3 


3 


3 


Cd 


3 


3 


3 


3 


cD 


2 






2 


cd 



Fig. 374. 



§ 894.] 



THE FOUR HUNTERS PROBLEM, 



505 



Now, if C and D never fished with A and B, or without A or 
B, then the combinations ABCD, abCD are inconsistent, and 
we eliminate them by making a figure 1 in those sections. 

Again, if when C and D stayed in, either A or B stayed in, 
then the combinations ABcd, abed, are inconsistent, and we 
eliminate them by making a figure 2 in those sections. 

Again, if C and D fished together or stayed in together, then 
all the combinations containing Cd, and also those containing 
cD, are inconsistent, and we eliminate them by making a figure 
3 in those sections. 

The uneliminated combinations are, omitting the small let- 
ters, ACD, BCD, A, B. 

These combinations show who went a-fishing on the four 
different occasions. 

894. Let us suppose the following state of facts: 

A party of four hunters, whom we will call A, B, C, and D, 
went into the woods to hunt deer. One or more went out hunt- 
ing on seven different days. A and C always went out 
together or stayed in together, and the same ones never went 
out twice, that is, a different party went out hunting each day. 

The problem is to tell who went together and who went alone 
on the seven different days. 

Let capital letters represent those who went out hunting. 

Let small letters represent the men who stayed in camp. 

Make an ABCD diagram: 



AB 


Ab 


aB 


ab 








1 


1 


CD 






1 


1 


Cd 


1 


1 






cD 


1 


1 






cd 



Fig. 375. 
Now, if A and C always went out together, or stayed in 
together, then all the combinations containing Ac, aC, are 



506 EXAMPLES. [Chap. 31. 

inconsistent, and we eliminate them by making a figure 1 in 
those sections. 

The uneliminated combinations are, omitting the small let- 
ters, ABCD, ABC, ACD, AC, BD, B, D. 

These combinations show who went out hunting each day. 

« 

895. Suppose the following state of facts: 

A common council was composed of five members, whom we 
will call A, B, C, D, and E. 

At a certain meeting of the council when they were all pres- 
ent, eleven roll calls were had on motions and the roll calls 
showed, 

(1) When A and C voted No, E voted Aye' with either B or D. 

(2) When A and B voted Aye and E voted No, B and C both 
voted Aye or both voted No. 

(3) When A and B, or A and E, or A, B, and E, voted Aye, C 
voted Aye and D voted No, or C voted No and D voted Aye, and 
conversely. 

Tell which four voted Aye on three ballots. 
Tell how A, B, and C each voted. 
Let capital letters represent those voting Aye. 
Let small letters represent those voting No. 
The premises can be stated thus : 

(1) ac = acBdE | acbDE, 

(2) ADe = ADeBC | ADebc, 

(3) AB | AE | ABE == Cd | cD, and conversely. 



§ 895.] THE COMMON COUNCIL PEOBLEM. 

Make an ABODE diagram: 



607 



AB 


Ab 


aB 


ab 




3 


3 






CDE 


3 


2 






CDe 






3 


3 


CdE 




3 


3 


3 


de 






1 :• 


3 


cDE 


2 


3 


31 


31 


i-De 


o 


3 




1 


cdE 


3 




1 


1 


ede 



Fig. 376. 

Now, if ac = BdE | bDE, then the combinations aBcde, 
abode, aBcDe, abcDe, abcdE, aBcDE, are inconsistent, and we 
eliminate them by making a figure 1 in those sections. 

Again, if ADe = BC | be, then the combinations AbCDe, 
ABcDe, are inconsistent, and we eliminate them bj making a 
figure 2 in those sections. 

Again, if AB | AE | ABE == Cd | cD, and conversely. If 
Cd | cD = AB | AE | ABE, then the following combina- 
tions ABcde, ABCDe* ABcdE, ABCDE, AbcdE, AbCDE, 
AbCde, AbcDe, aBCde, aBcDe, aBCdE, aBcDE, abCde, abcDe, 
abCdE, abcDE, are inconsistent, and we eliminate them by 
making a figure 3 in those sections. ^ . 

The uneliminated combinations are, 

Abcde, AbCdE, AbcDE, ABCdE, ABcDE, ABCde, abCDe, 
abCDfi, aBCDe, aBcdE, aBCDE. 

These combinations show that A, B, C, and E voted Aye 
together once, and B, C, D, and E voted Aye together once, 
and A, B, D, and E voted Aye together -once, . - 

A voted with C or D, except when B, C, and D voted, No. 



508 



EXAMPLES. 



[ Chap. 31. 



C voted with A or D, 
D voted with A or C. 
896. Let us take the Tenth Amendment to the Constitution 
of the United States, which reads as follows: 

The powers not delegated to the United States by the Con- 
stitution, nor prohibited by it to the States, are reserved to the 
States respectively, or to the people, and ascertain its latent 
meanings. 

Let a == the powers not delegated to the United States, 
b = the powers not prohibited to the states, 
C = the powers reserved to the states, 
D = the powers reserved to the people. 
The propositions contained in the amendment may be stated 
thus: 

(1) ab = Cd | cD 

(2) Cd | cD = ab 

(3) C = Cd 
Make an ABCD diagram: 



AB 


Ab 


aB 


ab 




3 


3 


3 


31 


I D 


2 


2 


2 




Cd 


2 


2 


2 




cD 








1 


cd 



Fig. 377. 

Now, if ab ^ Cd | cD, then the combinations abCD, abed, are 
inconsistent, and we eliminate them by making a figure 1 in 
those sections. 

Again, if Cd | cD «= ab, then the combinations ABCd, 
AbCd, aBCd, ABcD, AbcD, aBcD, are inconsistent, and we 
eliminate them by making a figure 2 in those sections. 

Again, if C = Cd, then the combinations ABCD, AbCD, 
aBCD, abCD, are inconsistent, and we eliminate them by mak- 
ing a figure 3 in those sections. -„^...v.^.-. ^, 



§ 896.] THE TENTH AMENDMENT. 509 

We can now obtain the following definitions from the Rea- 
soning Frame: 

(1) AB — ABcd, which we can translate thus: 

The powers delegated to the United States by the Consti- 
tution and prohibited by it to the States, are not reserved to the 
States, and are not reserved to the people. 

(2) Ab = Abed, which we can translate: 

The powers delegated to the United States by the Constitu- 
tion and not prohibited by it to the States, are not reserved to 
the States and are not reserved to the people. 

(3) C == abd, which we can translate: 

The powers reserved to the States are not delegated to the 
United States, and are not prohibited to the States and are not 
reserved to the people. 

(4) D — abc, which we can translate: 

The powers reserved to the people are not delegated to the 
United States, and are not prohibited to the States, and are 
not reserved to the States. 

(5) B = Be, which we can translate: 

The powers prohibited to the States are not reserved to the 
States. 

(6) No A = C, which can be translated : 

No powers delegated to the United States are reserved to the 
States. 

(7) No A = D, which can be translated: 

No powers delegated to the United States are reserved to 
the people. 

(8) No C = D, which can be translated: 

No powers reserved to the States are reserved to the people. 

The second definition which reads, 

The powers delegated to the United States and not pro- 
hibited to the States, are not reserved to the States, and are not 
reserved to the people, is a very important latent meaning of 
the amendment. 

The spirit of it seems to me to conflict with the spirit of cer- 
tain decisions of the supreme court of the United States where 
it has been held. 



510 



EXAMPLES. 



[ Chap. 31. 



(1) In the absence of congressional legislation on the sub- 
ject of bankruptcies, the States may pass insolvent laws, if 
they do not violate the obligation of contracts. It is not the 
mere existence of the power, but its exercise, which is incom- 
patible with the existence of the same power by the States. 

(2) The power to fix the standard of weights and measures 
is exclusive in Congress when exercised. 

(3) The power of Congress is not exclusive to provide for the 
punishment of counterfeiting the securities and current coin 
of the United States, etc. 

The extraordinary power of our system which is exemplified 
above, to develop every latent meaning of a clause in a consti- 
tution, statute, ordinance, will, contract, etc., ought to render 
it: very useful to judges and lawyers. 

With one operation it will tell us everything which a clause 
affirms, everything which it denies and everything which it 
leaves in doubt. 

Some of the examples already given show its remarkable 
ability to interpret the implied meanings contained in given 
statements of facts. This power ought to be very useful in 
cases depending on circumstantial evidence. 

897. This example is from Prof. Jevons : 

Where A is present, B and C are either both present at once, 
or absent at once, and where C is present A is present. 

Describe the class not-B under these conditions. 

The premises can be stated thus : 

(1) A = BC | bo 

(2) C = CA 
Make an ABC diagram: 



AB 


Ab 


aB 


ab 






i 


2 


2 


C 


1 






c 



Fig. 378. 



§ 893.] 



EXAMPLES FROM JEVONS. 



511 



Now, if A = BO | be, then the combinations ABc, AbC, are 
inconsistent, and we eliminate them by making a figure 1 in 
those sections. 

Again, if C = A, then the combinations containing aC are 
inconsistent, and we eliminate them by making a figure 2 in 
those sections. 

The Seasoning Frame now shows us that, 

b = c 

898. This example is from Prof. Jevons: 

It is known of certain things that, 

(1) Where the quality A is, B is not, 

(2) Where B is, and only where B is, C and D are. 
Derive from these conditions a description of the class of 

things in which A is not present but C is. 
The premises can be stated as follows: 

(1) A = Ab 

(2) B = CD 

(3) CD == B 
Make an ABCD diagram: 



AB 


Ab 


aB 


ab 




1 


3 




3 


CD 


12 




2 




Cd 


12 




2 




cD 


12 




2 




cd 



Fig. 379. 

Now, if A = b, then the combinations containing AB are 
inconsistent, and we eliminate them by making a figure 1 in 
those sections. 

Again, if B = CD, then all the B combinations, excepting 
those containing CD, are inconsistent, and we eliminate them 
by making a figure 2 in those sections. 

Again, if CD = B, then the combinations AbCD, abCD, are 



512 



EXAMPLES. 



[ Chap. 31. 



inconsistent, and we eliminate them by making* a figure 3 in 
those sections. 

The Reasoning Frame now shows that the definition of aO 
is, 

aC = BD | bd 

899. Taking the same premises as in the previous section, 
draw descriptions of the classes Ac, ab and cD. 

The Reasoning Frame shows that the definitions are, 

(1) Ac = Acb 

(2) ab = Cd | cD | cd 

(3) cD = cDb 

900. The following example is from Prof. DeMorgan: 
Every A is one only of the two, B or C ; D is both B and C, 

except when D is E, and then it is neither, therefore. 

No A is D 
The premises can be stated thus: 

(1) A == Be | bC 

(2) D == BCe | bcE 
Make an ABCDE diagram: 



AB 


Ab 


bB 


ab 




1 


2 


2 


2 


CUE 


1 


2 




2 


CDe 


1 








CdE 


1 




2 




Cdf 


2 


1 




cDE 


2 


1 


2 


2 


c-De 




1 






cdE 




1 






cde 



Fig. 380. 
Now, if A = Be | bC, then the combinations containing 



901.] 



PROBLEM ABOUT A CLASS OF THINGS. 



513 



ABC, Abe, are inconsistent, and we eliminate them by making 
a figure 1 in those sections. 

Again, if D = BCe | bcE, then in the remaining combina- 
tions, all the D combinations, excepting aBCDe, abcDE, are 
inconsistent and we eliminate them by making a figure 2 in 
those sections. 

The Seasoning Frame now shows that, 

No A = D 

901. The following example is taken from * * * (the 
lettering is mine) : 

"There is a certain class of things (D) from which E picks 
out the A that is B and the C that is not B, and F picks out 
from the remainder the B which is C and the A that is not C. 
It is then found that nothing is left but the class B which is not 
A. The whole of this class is, however, left. What can be 
determined about the class originally? 

Make an ABCD diagram : 



AB 


Ab 


aB 


ab 




1 


2 


3 


2 


CD 










Cd 


1 


4 






cD 










cd 



Fig. 381. 

Now, if E picks out of the D things, the A that is B, then we 
eliminate the combinations containing DAB by making a figure 
1 in those sections. 

Again, if E picks out of the D things the C that is b, then we 
eliminate the combinations OAbD, CabD, by making a figure 2 
in those sections. 

Again, if F picks out from the remainder of the D things the 
B which is C, then we eliminate the DaBC combination by mak- 
ing a figure 3 in that section. 

Again, if F picks out of the remainder of the D things the A 

33 



514 EXAMPLES. [ Chap. 31. 

which is c, then we eliminate the combination DAbc by making 
a figure 4 in that section. 

The Reasoning Frame now shows that there are two D com- 
binations left, viz. : aBcD, abcD. 

This proves that the answer given in the text is wrong. 
Originally the class D contained the following classes of 
things: ABC, ABc, AbC, Abe, aBC, aBc, abC, abc. 

902. The following example is from Dr. Keynes, p. 433: 

•'Show what may be inferred as a possible description of 
warm-blooded vertebrates from the following, and state 
whether any of the information there given is superfluous for 
the purpose: 

(1) All vertebrates may be divided into warm-blooded and 
cold-blooded, and all produce their young in but one of the two 
ways, i. e., are either viviparous or oviparous. 

(2) No feathered vertebrate is both viviparous and warm- 
blooded. 

(3) No oviparous vertebrate that is cold-blooded has 
feathers. 

(4) Every viviparous vertebrate is either feathered or warm- 
blooded. 

Let A == vertebrates, 

B = warm-blooded, 

C == cold-blooded, 

D = viviparous, 

E = oviparous, 

F — feathered. 
The premises can be stated thus : 

(1) A = Be | bC 

(2) No B = C 

(3) A = De | dE 

(4) No D = E 

(5) No AF == BD 

(6) No ACE == F 

(7) AD = bF | Bf 



§ 902.] PROBLEM ABOUT WARM-BLOODED VERTEBRATES. 515 
Make an ABCDEF diagram: 



ABC 


ABc 


AbC 


Abe 


aBC 


aBc 


abC 


abc 




536 
21 

74 


35 

47 


36 
4 


3 
41 


42 


4 


4 


4 


DEF 


21 
43 


34 


3 

74 


7 
314 


42 


4 


4 


4 


DEf 


21 

7 5 


57 




1 


2 








DeF 


21 




7 


17 


2 








Def 


216 




6 


1 


2 








dEF 


21 






1 


2 








dEf 


213 


3 


3 


31 


2 








deF 


213 


3 


3 


31 


2 








def 



Fig. 382. 

Now, if A = Be | bC, then all the A combinations containing 
BC and be are inconsistent, and we eliminate them by making 
a figure 1 in those sections. 

Again, if No B = C, then all the combinations containing BC 
are inconsistent, and we eliminate them by making a figure 2 in 
those sections. 

Again, if A = De | dE, then all the combinations containing 
ADE, Ade, are inconsistent, and we eliminate them by making 
a figure 3 in those sections. 

Again, if No D = E, then all the DE combinations are incon- 
sistent, and we eliminate them by making a figure 4 in those 
sections. 

Again, if No AF = BD, then the combinations containing 
ABDF are inconsistent, and we eliminate them by making a 
figure 5 in those sections. 

Again, if No ACE = F, then the combinations containing 
ACEF are inconsistent, and we eliminate them by making a 
figure 6 in those sections. 

Again, if AD = bF | Bf, then the combinations ABDF, 



516 EXAMPLES. [ Chap. 31. 

ADbf, are inconsistent, and we eliminate them by making a 
figure 7 in those sections. 

The Seasoning Frame now gives us the following definition 
of warm-blooded vertebrates: 

AB = cDef | cdEF | cdEf, which can be translated, 

Warm-blooded vertebrates are viviparous, and featherless or 
oviparous. 

1 think that the information, No feathered vertebrate is both 
viviparous and warm-blooded is superfluous, for the reason that 
the combinations which it causes us to eliminate are eliminated 
by one or more of the other premises. 

The reasoning Frame also gives us this definition: 
AC == bDeF | bDef | bdEf, 
which can be translated, 

Cold-blooded vertebrates are viviparous and feathered, or 
oviparous and featherless. 

903. (1) In a certain town the old buildings are either 
ecclesiastical and built entirely of stone, or, if not ecclesiasti- 
cal, are built entirely of brick. 

(2) The brick and stone buildings are all modern as well as 
secular, or they are neither. 

(3) But there are no modern buildings at once secular and 
built entirely of stone. 

State what assumptions you make in interpreting the above 
and determine, 

(a) In what cases brick may be found in the buildings of this 
town and in what cases it cannot be. 

(b) What old buildings it would be useless to look for." 

(Keynes Formal Logic, p. 434). 
Let A == buildings, 
B — ecclesiastical, 
C = built of stone, 
D = built of brick, 
E = modern, 
F = secular, 
G = old. 



§ 903.] 



PROBLEM ABOUT BUILDINGS. 



517 



The premises can be stated thus: 

(1) AG = BCd | bcD, 

(2) ACD == EF | ef 

(3) Xo AE — CF 

(4) No E = G 

(5) Xo B = F 

(6) Nob = f 

(7) Xoe =g 
Make an ABCDEFG diagram: 



ABC 


ABc 


AbC 


Abe 


aBC 


aBc 


abC 


abc 




5134 


514 


341 


4 


54 


54 


4 


4 


DEFG 


53 

241 


5 


3 




5 


5 






DEFg 


41 


2416 


64 


4 


4 


64 


64 


DEfG 


o 




62 


6 






6 


6 


DEfg 


521 


51 


21 




5 


5 






DeFG 


752 


7 5 


72 


7 


7 5 


7 5 


7 


' 


DeFg 


1 


1 


61 


6 






6 


6 


DefG 


7 


7 


76 


76 


7 


7 


76 


76 


Defg 


543 


54 1 


41 


431 


54 


5 4 


4 


4 


dEFG 


53 


5 


3 




5 


5 






dEFg 


4 


41 


G41 


641 


4 


4 


64 


64 


dEG 






6 


6 






6 


6 


dEfg 


5 


5 1 


1 


1 


5 


5 






deFG 


7 5 


7 5 


7 


7 


75 


7 5 


7 


7 


deFg 




1 


61 


61 






6 


6 


defG 


7 


7 


76 


76 


7 


7 


76 


76 


defg 



Fig. 383. 



518 EXAMPLES. [ Chap. 31. 

Now, if AG = BCd | bcD, then all the combinations con- 
taining AGBc, AGbC, AGBCD, AGbcd, are inconsistent, and 
we eliminate them by making a figure 1 in those sections. 

Again, if ACD = EF | ef, then the combinations containing 
ACDEf, ACDeF, are inconsistent, and we eliminate them by 
making a figure 2 in those sections. 

Again, if No AE = OF, then the combinations containing 
AECF are inconsistent, and we eliminate them by making a 
figure 3 in those sections. 

Again, if No E = G, then all the combinations containing 
EG are inconsistent, and we eliminate them by making a figure 

4 in those sections. 

Again, if No B = F, then all the combinations containing 
BF are inconsistent, and we eliminate them by making a figure 

5 in those sections. 

Again, if No b = f , then the combinations containing bf are 
inconsistent, and we eliminate them by making a figure 6 in 
those sections. 

i^gain, if No e = g, then the combinations containing eg are 
inconsistent, and we eliminate them by making a figure 7 in 
those sections. 

We have made the following assumptions, viz : 

(1) Old an<J modern, not old and not modern, ecclesiastical 
and secular, not ecclesiastical and not secular, are inconsistent 
combinations. 

The Reasoning Frame now gives us the following defini- 
tions : 

AD = BcEfg | bcEFg | bceFG, 
which can be translated, 

The brick buildings are ecclesiastical and modern, or modern 
and secular, or secular and old. 

No D == ABG, 
which can be translated, 

No brick can be found in old ecclesiastical buildings. 

904. (1) If a nation has natural resources and a good gov- 
ernment, it will be prosperous. 



8 904.] 



PROBLEM ABOUT A NATION. 



519 



(2) If it has natural resources without a good government, 
or a good government without natural resources, it will be 
contented but not prosperous. 

(3) If it has neither natural resources nor a good govern- 
ment, it will be neither contented nor prosperous. 

Show that these statements may be reduced to two propo- 
sitions of the form of Hamilton's TJ. 
Let A = nation 

B = natural resources 
C = good government 
D = prosperous 
E = contented. 
The premises can be stated as follows : 

(1) ABC = ABCD 

(2) ABc | AbC = AdE 

(3) Abe — Ade 
Make an ABODE diagram: 



AB 


Ab 


aB 


ab 






2 




CDE 




2 






De 


1 








CdE 


1 


9 






Cde 


o 


3 






cDE 


2 


3 






cDe 




3 






cdE 


2 








cde 



Fig. 384. 

Now, if ABC = ABCD, then the combinations containing 
ABCd are inconsistent and we eliminate them by making a 
figure 1 in those sections. 



520 EXAMPLES. [Chap. 31. 

Again, if ABc | AbC == AdE, then the combinations 
ABcDE, ABcDe, ABcde, AbCDE, AbCDe, AbCde, are incon- 
sistent and we eliminate them by making a figure 2 in those 
sections. 

Again, if Abe = Ade, then the combinations AbcDE, 
AbcDe, AbcdE, are inconsistent and we eliminate them by 
making a figure 3 in those sections. 

This Seasoning Frame now shows the logical expression of 
the given premises but I think that the premises cannot be 
reduced to two propositions of the form of Hamilton's U. 

We can get ABO == AD (1); Abe = Ade (2); and AbC | 
ABc = AdE (3) ; but it will be seen that (3) prevents us from 
reducing the statements to two IT propositions. 

905. Let the observation of a class of natural productions 
be supposed to have led to the following general results: 

(1) That in whichsoever of these productions the properties 
A and C are missing, the property E is found, together 
with one of the properties B and D, but not with both. 

(2) That wherever the properties A and D are found, while 
E is missing, the properties B and C will either both be 
found or both be missing. 

(3) That wherever the property A is found in conjunction 
with either B or E, or both of them, there, either the prop- 
erty C or the property D will be found, but not both of 
them. 

(4) Wherever the property G or D is found, there the prop- 
erty A will be found in conjunction with either B or E, 
or both of them. 

The premises can be stated as follows: 

(1) ac = BdE | bDE 

(2) ADe = BC | be 

(3) ABE | AbE | ABe = Cd | cD 

(4) Cd I cP = ABe 1 AbE | ABE. 



g 905.] A CLASS OF NATURAL PRODUCTIONS. 

Make an ABCDE diagram: 



521 



AB 


Ab 


aB 


ab 




3 


3 






CDE 


3 


2 






CDe 






4 


4 


CdE 




4 


4 


4 


Cde 






41 


4 


cDE 


~ 


4 


4 

1 


4 

1 


cDe 


3 


3 




1 


cdE 


3 




1 


1 


cde 



Fig. 385. 

Now, if ac = BdE | bDE, then the combinations acBDE, 
aeBDe, acBde, acbDe, abcdE, acbde, are inconsistent, and we 
eliminate them by making a figure 1 in those sections. 

Again, if ADe = BC | be, then the combinations ABcDe, 
AbCDe, are inconsistent and we eliminate them by making a 
figure 2 in those sections. 

Again, if ABe | AbE | ABE = Cd | cD, then the combi- 
nations ABCDE, ABCDe, ABcdE, ABcde, AbCDE, AbcdE, 
are inconsistent and we eliminate them by making a figure 3 
in those sections. 

Again, if Cd | cD = ABe | AbE | ABE, then the combina- 
tions AbCde, AbcDe, aBCdE, aBCde, aBcDE, aBcDe, abCdE, 
abCde, abcDE, abcDe, are inconsistent and we eliminate them 
by making a figure 4 in those sections. 

We can now get the following definition of A : 

A = BCdE | BCde | BcDE | bCdE | bcDE | bede, 
and the following definition of aC: 

aC = D, 
and the following definition of Cd | Dc, 



522 EXAMPLES. [ Chap. 31. 

Cd | cD = ABE | ABe | AbE, 
and the following definition of bed, 

bed = Ae. 
These definitions show that where A is found, there also C oc 
D are found, or else B, C and D are absent, and where C or D is 
found, or B, C and D are together absent, A is found; and if 
A is absent and C present, D is present. 

906. Given the same premises as in the preceding section, 
show that, 

(1) If the property B be present in one of the productions, 
either the properties A, C and D are all absent, or some 
one alone of them is absent, and conversely, if they are 
all absent, it may be concluded that the property B is 
present. 

(2) If A and C are both present or both absent, D will be 
absent quite independently of the presence or absence 
of B. 

We can get the following definition of B: 

B == ACdE | ACde | AcDE | aCDE | aCDe | acdE. 
This definition shows that if B be present, A, C and D are all 
absent, or some one alone of them is absent, and it also 
shows that if A, C and D are all absent, B is present. 
We can also get the following definition of AC | ac: 

AC | ac = d. 
This shows that if A and C are both present, or both absent, 
D is also absent. 

907. The following is adapted from an example in Prof. 
Venn's "Symbolic Logic;" 

Given BD == A 
DE == C 
Find BE in terms of A and C. 



908.] PROBLEM FROM "SYMBOLIC LOGIC." 

Make an ABODE diagram: 



523 



AB 


Ab 


aB 


ab 








1 




CDE 






1 




CDe 










CdE 










Cde 


o 


- 


1 


g 


cDE 






1 




cDe 










cdE 










cde 



Fig. 386. 

Now, if BD = A, then the combinations containing BDa 
are inconsistent and we eliminate them by making a figure 1 
in those sections. 

Again, if DE = C, then the combinations containing DEc 
are inconsistent and we eliminate them by making a figure 2 
in those sections. 

We can now get the following definition of BE : 
BE = ACD | ACd | Acd | acd | aCd. 

From this we can get, 

BE = AC | cd | aCd. 

908. Are the three following systems of propositions equiv- 
alent? 

(1) Ab = cd 
aB = Ce 
D = E 

(2) A = B | c | D 
BE = A 

Be = Ad | Cd 
bD = aE 



524 



EXAMPLES. 



[ Chap. 31. 



(3) A | e -== B | d 

a = bE | bd | BCe 
be = a 
D == E. 
Make an ABODE diagram: 



AB 


Ab 


aB 


ab 






1 


2 




CDE 


3 


13 


3 


3 


CDe 




1 


2 




CdE 




1 






Cde 




1 


2 




cDE 


3 


13 


23 


3 


cDe 






2 




cdE 






2 




cde 



Fig. 387. 

Now, if Ab = cd, then the combinations containing AbCD, 
AbCd, AbcD are inconsistent and we eliminate them by making 
a figure .1 in those sections. 

Again, if aB = Ce, then the combinations containing aBE, 
aBc, are inconsistent and we eliminate them by making a 
figure 2 in those sections. 

Again, if D === E, then all the combinations containing De 
are inconsistent and we eliminate them by making a figure 3 
in those sections. 

This Reasoning Frame shows the logical expression of the 
first system of propositions. 



§ 908.] THREE SYSTEMS OF PROPOSITIONS. 

Make an ABCDE diagram: 



525 



AB 


Ab 


aB 


ab 




1 


4 


2 




CDE 


13 


4 


3 


4 


CDe 




1 


2 




CdE 


3 


1 






Cde 


1 


14 


2 




cDE 


13 


14 


3 


4 


cDe 


1 




2 




cdE 


1 




3 




cde 



Fig. 388. 

Now, if A = B | c | D, then the combinations containing 
ABCD, ABcD, ABcd, AbCd, AbcD, are inconsistent and we 
eliminate them by making a figure 1 in those sections. 

Again, if BE = A, then the combinations containing BEa, 
are inconsistent and we eliminate them by making a figure 
2 in those sections. 

Again, if Be — Ad | Cd, then the combinations ABCDe, 
ABCde, ABcDe, aBCDe, aBcDe, aBcde, are inconsistent and 
we eliminate them by making a figure 3 in those sections. 

Again, if bD= aE, then the combinations containing AbD, 
abDe, are inconsistent and we eliminate them by making a 
figure 4 in those sections. 

This Reasoning Frame shows the -logical expression of the 
second system of propositions and it also shows that the first 
and second systems are not equivalents. 



526 EXAMPLES. 

Make an ABCDE diagram: 



[ Chap. 31. 



AB 


Ab 


aB 


ab 






31 


2 




CDE 


4 


4 
3 


4 


4 
21 


CDe 


1 


3 


2 


2 


CdE 




3 


I 




Cde 




1 


2 




cDE 


4 


4 


4 
2 


4 
21 


cDe 


1 




2 


2 


cdE 






21 




cde 



Fig. 389. 

Now, if A | e = B | d, then the combinations containing 
ABdE, AbDE, aBde, abDe, are inconsistent and we eliminate 
them by making a figure 1 in those sections. 

Again, if a = bE | bd | BCe, then the following combina- 
tions containing aBCE, aBc, abCDe, abCdE, abcDe, abcdE, 
are inconsistent and we eliminate them by making a figure 2 
in those sections. 

Again, if bC = a, then the combinations containing bCA are 
inconsistent and we eliminate them by making a figure 3 in 
those sections. 

Again, if D = E, then all the combinations containing De 
are inconsistent and we eliminate them by making a figure 4 
in those sections. 

This Keasoning Frame shows the logical expression of the 
third system of propositions and it also shows that it is not 
equivalent to either one of the first and second systems. 

909. The following example is adapted from Dr. Keynes' 
"Formal Logic," p. 438. 

(1) Abe = DE 

(2) B = C 



§ 909.] 



PROBLEM FROM "FORMAL LOGIC." 



527 



(3) b = c 

(4) BCD = AE | ae 

(5) BCd == Ae 

(6) abcD = E 
Then it follows that, 

(1) Ab = cDE 

(2) Ad = BCe 

(3) aB = CDe 

(4) aE = be 

(5) Bd = ACe 

(6) BE = ACD 

(7) bd = cE 

(8) bd = ae 

(9) be = acd 
(10) dE = abc. 

Make an ABODE diagram: 



AB 


Ab 


aB 


ab 






3 


4 


3 


CDE 


4 


3 




3 


CDe 
CdE 


5 


3 


5 


3 




3 


5 


3 


Cde 


o 




2 




cDE 


2 


1 


2 


6 


cDe 


•> 


1 


2 




cdE 


o 


1 


2 




cde 



Fig. 390. 

Now, if Abc = DE, then the combinations containing 
AbcDe, AbcdE, Abcde, are inconsistent and we eliminate them 
by making a figure 1 in those sections. 

Again, if B = C, then the combinations containing Be are 



528 EXAMPLES. [ Chap. 31. 

inconsistent and we eliminate them by making a figure 2 in 
those sections. 

Again, if b = c, then the combinations containing bC are 
inconsistent and we eliminate them by making a figure 3 in 
those sections. 

Again, if BCD = AE | ae, then the combinations ABCDe, 
aBCDE, are inconsistent and we eliminate them by making 
a figure 4 in those sections. 

Again, if BCd = Ae, then the combinations ABCdE, aBCdE 
aBCde, are inconsistent and we eliminate them by making a 
figure 5 in those sections. 

Again, if abcD == E, then the combination abcDe is incon- 
sistent and we eliminate it by making a figure 6 in that section. 

We can now get the following definitions : 

(1) Ab == cDE 

(2) Ad == BCe 

(3) aB = CDe 

(4) aE = be 

(5) Bd = ACe 

(6) BE = ACD 

(7) bD = cE 

(8) bd == ac 

(9) be = acd 
(10) dE = abc. 

910. The members of a scientific society are divided into 
three sections, which are denoted by A, B, C. Every mem- 
ber must join one at least, of these sections, subject to the 
following conditions: 

(1) Any one who is a member of A but not of B, of B but not 
of C, or of C but not of A, may deliver a lecture to the 
members if he has paid his subscription, but, otherwise, 
not. 

(2) Any one who is a member of A, but not of C, of C but not 
of A, or of B but not of A, may exhibit an experiment 
to the members if he has paid his subscription, but, other- 
wise, not. 



§ 910.] PROBLEM ABOUT THE MEMBERS OF A SOCIETY. 529 

(3) But every member must either deliver a lecture or per- 
form an experiment annually, before the other members. 
Find the least addition to these rules which will compel every 
member to pay his subscription or forfeit his membership. 
Let D = members of the society, 
A = member of section A, 
B = member of section B, 
C = member of section C, 
E = one who delivers a lecture, 
F = one who has paid his subscription, 
G = one who performs an experiment. 
The premises are, 

(1) Abe | aBc | abC = EF | ef 

(2) Abe | aBc | abC *= FG | fg 

(3) D == E | O 

(4) No A = d 

(5) No B = d 

(6) No C = d. 



34 



530 EXAMPLES. 

Make an ABCDEFG diagram: 



[ Chap. 31. 



ABC 


ABc 


AbC 


Abe 


aBC 


aBc 


abC 


abc 




3 


3 


3 


3 


3 


3 


3 


3 


DEFG 








2 




2 


2 




DEFg 


3 


3 


3 


312 


3 


231 


132 


3 


DEfG 








1 




1 


1 




DEfg 








1 




1 


1 




DeFG 


3 


3 


3 


312 


3 


123 


123 


3 


DeFg 








2 




2 


2 




DefG 


3 


3 


3 


3 


3 


3 


3 


3 


Defg 


546 


54 


64 


4 


65 


5 


6 




dEFG 


546 


54 


64 


42 


65 


52 


62 




dEFg 


546 


54' 


64 


412 


65 


521 


261 




dEfG 


546 


54 


64 


41 


65 


51 


61 




dEfg 


546 


54 


64 


41 


65 


51 


61 




deFG 


546 


54 


64 


412 


65 


512 


612 




deFg 


546 


54 


64 


42 


65 


5 2 


62 




defG 


546 


54 


64 


4 


65 


5 


6 




defg 



Fig. 391. 

Now, if Abe | aBc | abC == EF | ef , then the combinations 
containing AbcEf, AbceF, aBcEf, aBceF, abCEf, abCeF, are 
inconsistent and we eliminate them by making a figure 1 in 
those sections. 

Again, if Abc | aBc | abC = FG | fg, then the combina- 
tions containing AbcFg, AbcfG, aBcFg, aBcfG, abCFg, abCfG, 



§911.] "HE THAT BELIEVETH, ETC." 531 

are inconsistent and we eliminate them by making a figure 2 
in those sections. 

Again, if D = E | G, then the combinations containing 
DEG, Deg, are inconsistent and we eliminate them'by making 
a figure 3 in those sections. 

Again, if No A = d, then all the combinations containing 
Ad are inconsistent and we eliminate them by making a figure 

4 in those sections. 

Again, if No B = d, then all the combinations containing 
Bd are inconsistent and we eliminate them by making a figure 

5 in those sections. 

Again, if No C = d, then all the combinations containing 
Cd, are inconsistent and we eliminate them by making a figure 

6 in those sections. 

The Seasoning Frame now shows that if we add either one of 
the following rules, every member must pay his subscription 
or forfeit his membership: 

(1) D = F, which can be translated, 

Every member is one who has paid his subscription. 

(2) No D = f , which can be translated, 

No one is a member who has not paid his subscription. 

(3) No f = E | G, which can be translated, 

No one who has not paid his subscription can deliver a 
lecture or perform an experiment annually before the 
other members. 
The first one seems to make the least addition to the rules. 
911. Let us take this example: 

"He that believeth and is baptized shall be saved but he 
that believeth not shall be damned." 
I assume that those who shall be saved shall not be damned. 
Let A = those who believe, 

B = those who are baptized, 
C = saved, 
D = damned. 



532 



EXAMPLES. 



[ Chap. 31. 



The premises can be stated thus : 

(1) AB = ABC 

(2) a = aD 

(3) No C = D 
Make an A BCD diagram : 



AB 


Ab 


aB 


ab 




3 


3 


3 


3 


CD 






2 


2 


Cd 


1 








cD 


1 




2 


2 


cd 



Fig. 392. 

Now, if AB = ABC, then the combinations containing ABc, 
are inconsistent and we eliminate them by making a figure 1 
in those sections. 

Again, if a = aD. then the combinations containing ad are 
inconsistent and we eliminate them by making a figure 2 in 
those sections. 

Again, if No C = D. then the combinations containing CD 
are inconsistent and we eliminate them by making a figure 
3 in those sections. 

From the uneliminated combinations we can now get the 
following consistent definitions: 

(1) AB = ABC 
which can be translated, 

Those who believe and are baptized shall be saved. 

(2) Ab = Cd | cD | cd, 
which can be translated, 

Those who believe and are not baptized shall be saved 
or damned or neither. 

(3) C = AB | Ab, 
which may be translated, 

Those who shall be saved are believers, and baptized or 
not baptized. 



§912.] "EXCEPT A MAN BE BORN OF WATER, ETC." 533 

(4) D = Ab | aB | ab, 
which can be translated, 

Those who shall be damned are either believers who are 
not baptized or not believers, whether baptized or not 
baptized. 

(5) a = D, 
which can be translated, 

Those who do not believe shall be damned. 

(6) c = D | a, 
which can be translated, 

Those who shall not be saved shall be damned, or not 
damned. 

(7) d = AB | Ab, 
which can be translated, 

Those who shall not be damned are either those who 
believe and are baptized or those who believe and are 
not baptized. 
912. Let us take this example. 

"Except a man be born of water and of the spirit he 
cannot enter into the kingdom of God." 
The logical meaning of this passage is, 

If a man is not born of water and of the spirit he cannot 

enter into the kingdom of God. 
Let A == man, 

B = born of water, 
= born of the spirit, 
D === enter into the kingdom of God. 
(1) If A — be, then A = d, 
which can be reduced to, 

Abe = Abed. 



534 EXAMPLES. 

Make an ABCD diagram : 



[ Chap. 31. 



AB 


Ab 


aB 


ab 










• 


CD 










Cd 




1 






cD 










cd 



Fig. 393. 

Now, if Abe = Abed, then the combination AbcD is incon- 
sistent and we eliminate it by making a figure 1 in that section. 

From the uneliminated combinations we can get the follow- 
ing consistent definitions: 

(1) Abe = d, 
which can be translated, 

A man not born of water and of the spirit shall not enter 
into the kingdom of God. 

(2) ABC = D | d, 
which can be translated, 

A man born of water and of the spirit shall or shall not 
enter into the kingdom of God. 

(3) ABc = D | d, 
which can be translated, 

A man born of water and not of the spirit, shall or shall 
not enter into the kingdom of God. 

(4) AbO = D | d, 
which can be translated, 

A man born of the spirit and not of water shall or shall 
not enter into the kingdom of God. 

(5) D = ABC | ABc | AbC | a, 
which can be translated, 

Those who shall enter into the kingdom of God are men 
born of water and of the spirit, or men born of water 
and not of the spirit, or men born of the spirit and 
not of water, or not-men. 



CHAPTER XXXII. 

INDUCTIVE EXAMPLES. 

913. Given the combinations ABC or Abe or aBC or abC, 
we are to find a set of propositions not involving alternative 
combinations which shall produce them. 

Make an ABC diagram: 



AB 


Ab 


aB 


ab 




s 


1 


S 


S 


c 


1 


s 


1 


1 


c 



Fig. 394. 

In the sections which contain the given alternants, viz.: 
ABC, Abe, aBC, abC, make a letter S to indicate that these 
combinations are saved, that is, they are not to be eliminated. 

The Reasoning Frame now shows us that the proposition, 

bC = a, will cause us to eliminate the combination AbC, 
and the proposition, 

c = Ab, will cause us to eliminate the combinations ABc, 
aBc, abc. 

Mark the eliminated combinations with a figure 1. The 
given combinations are all saved. This proves that the two 
propositions, 

bC = a 
c = Ab 
will produce the given combinations, ABC or Abc or aBC or 
abC. 

914. The given alternants are ACe, aBCe, aBcdE, abCe, 
abcE. 



536 INDUCTIVE EXAMPLES. 

Make an ABODE diagram: 



[Chap. 32. 



AB 


Ab 


aB 


ab 












CDE 


S 


• 
S 


S 


S 


CDe 










CdE 


s 


s 


s 


s 


Cde 








s 


cDE 










cDe 






s 


IS 


cdE 










cde 



Fig. 395. 

Now make a letter S in the sections containing the given 
alternants. 

By reading the definitions which we can get from the combi- 
nations which are in the sections marked with the letter S, we 
can get the following definitions: 

(1) A == CeA 

(2) BD = CeBD 

(3) e = C 

(4) C = e 

These definitions will produce the combinations ACe | aBCe 
I aBcdE I abCe I abcE. 



§ 914.] AN EXAMPLE. 

Make an ABCDE diagram: 



537 



AB 


Ab 


aB 


ab 




4 
12 


4 

1 


4 
2 


4 


CDE 










CDe 


4 
1 


4 
1 


4 


4 


CdE 










Cde 


12 


1 


2 




cDE 


3 
12 


3 

1 


3 
2 


3 


cDe 


1 


1 






cdE 


31 


31 


3 


3 


cde 



Fig. 396. 

Now, if A = Ce, then the combinations containing ACE, 
AcE, Ace, are inconsistent, and we eliminate them by making 
a figure 1 in those sections. 

Again, if BD = Ce, then the combinations containing BDCE, 
BDcE, BDce, are inconsistent, and we eliminate them by mak- 
ing a figure 2 in those sections. 

Again, if e = C, then the combinations containing ec are 
inconsistent, and we eliminate them by making a figure 3 in 
those sections. 

Again, if C = e, then the combinations containing CE are 
inconsistent, and we eliminate them by making a figure 4 in 
those sections. 

An examination of the two Reasoning Frames now shows us 
that the propositions, 

A = CeA 
BD = CeBD 
e = C 
C = e 
will produce the given combinations. 



538 



INDUCTIVE EXAMPLES. 



[ Chap. 32. 



The propositions which we have given are not the only prop- 
ositions which will produce the given combinations. 

We could find several sets of propositions which taken 
together would produce the given combinations. 

We can proceed in a tentative manner by taking one defini- 
tion which the given combinations will yield, as a new premise, 
and then proceed to eliminate the combinations which are 
inconsistent with it. Then get another definition from the 
given combinations and eliminate the inconsistent combina- 
tions. And so continue this process of getting definitions 
from the given combinations and eliminating the inconsistent 
combinations, until all the inconsistent combinations are elim- 
inated, being careful not to get any definition from the given 
combinations which would cause the elimination of any alter- 
nant in the given combinations. 

915. The given combinations are ABCD | ABCd | ABcd 
| AbCD | AbcD | aBOD | aBcD | aBcd | abCd. 

Make an ABCD diagram : 



AB 


Ab 


aB 


ab 




S 


S 


S 




CD 


S 






S 


Cd 




s 


s 




cD 


s 




s 




cd 



Fig. 397. 

Mark the given alternants with the letter S. 

It will be understood by the reader that all the combinations 
not marked with the letter S are eliminated. 

An examination of the Seasoning Frame shows that we cau 
get the following definitions: 

(1) cd == Bed 

(2) ab = Cdab 

(3) AbC = DAbC 

(4) aBC = DaBC 

(5) ABD = CABD 



§ 916.] AN EXAMPLE. 

Make an ABCD diagram: 



539 



AB 


Ab 


aB 


ab 










2 


CD 




3 


4 




Cd 


5 






2 


cD 




1 




21 


cd 



Fig. 398. 

Now, if cd = B, then the combinations containing Acd, abed, 
are inconsistent, and we eliminate them by making a figure 1 
in those sections. 

Again, if ab === Cd, then the combinations containing abc, 
abCD, are inconsistent, and we eliminate them by making a 
figure 2 in those sections. 

Again, if AbC = D, then the combination AbCd is incon- 
sistent, and we eliminate it by making a figure 3 in that section. 

Again, if aBC = D, then the combination aBCd is inconsis- 
tent, and we eliminate it by making a figure 4 in that section. 

Again, if ABD = 0, then the combination ABcD is incon- 
sistent, and we eliminate it by making a figure 5 in that sec- 
tion. 

The result proves that we have found propositions which 
will produce the given combinations. 

916. The given combinations are ABCDE | ABCDe i 
ABCde | ABcde [ AbCDE | AbcdE | Abcde | aBCDe ! 
aBCde I aBcDe I abCDe I abCdE I abcDe I abcdE. 



540 INDUCTIVE EXAMPLES. 

Make an ABODE diagram : 



[Chap. 32. 



AB 


Ab 


aB 


ab 




S 


S 






CDE 


S 




S 


S 


CDe 








s 


CdE 


s 




s 




Cde 










cDE 






s 


s 


cDe 




s 




s 


cdE 


s 


s 






cde 



Fig. 399. 

Mark the sections containing the given alternants with a 
letter S. 

From the given alternants we can obtain the following defi- 
nitions : 

(1) AbC = DEAbO 

(2) ABc = deABc 

(3) aBc = DeaBc 

(4) CdE == abCdE 

(5) DE = ADE 

(6) Abe = dAbc 

(7) abCd = EabCd 

(8) abed = Eatcd 



916.] AN EXAMPLE. 

Make an ABODE diagram: 



541 



AB 


Ab 


aB 


ab 








5 


5 


CDE 




1 






CDe 


4 


41 


4 




CdE 




1 


7 


Cde 


2 


6 


5 
3 


5 


cDE 


2 


6 






cDe 


o 




3 




cdE 






3 


8 


cde 



Fig. 400. 

Now, if AbC = DE, then the combinations containing AbCd, 
AbCDe, are inconsistent, and we eliminate them by making a 
figure 1 in those sections. 

Again, if ABc = de, then the combinations containing 
ABcD, ABcdE, are inconsistent, and we eliminate them by 
making a figure 2 in those sections. 

Again, if aBc = De,then the combinations containing aBcd, 
aBcDE, are inconsistent, and we eliminate them by making a 
figure 3 in those sections. 

Again, if CdE = ab, then the combinations containing 
ACdE, aBCdE, are inconsistent, and we eliminate them by 
making a figure 4 in those sections. 

Again, if DE = A, then the combinations containing DEa 
are inconsistent, and we eliminate them by making a figure 5 
in those sections. 

Again, if Abe — d, then the combinations containing AbcD 
are inconsistent, and we eliminate them by making a figure (> 
in those sections. 



542 INDUCTIVE EXAMPLES. [ Chap. 32. 

Again, if abCd == E, then the combination containing abCde 
is inconsistent, and we eliminate it by making a figure 7 in that 
section. 

Again, if abed = E, then the combination containing abede 
is inconsistent, and we eliminate it by making a figure 8 in that 
section. 

The result proves that we have obtained a set of propositions 
which will produce the given combinations. Bythesame method 
we can obtain other sets of propositions which will produce 
the given combinations. 



CHAPTER XXXIII. 



EQUIVALENTS FOR PROPOSITIONS. 



917. By the use of the Reasoning Frame I have discovered 
a new method of finding a set of propositions which shall be 
equivalent to a complex categorical proposition, or to a com- 
bination of complex categorical propositions. 

The method is as follows: 

(1) State the given proposition, 

(2) Eliminate the inconsistent combinations, 

(3) From the consistent combinations which remain, get 

definitions of any letter-term and its negative. 
These definitions will be equivalent propositions to the given 
proposition, or, 

(4) From the consistent combinations get definitions by 

the method described in the preceding chapter. 
The* definitions thus obtained will make a set of 
propositions equivalent to the given proposition. 
Let the given propositions be, 
aB = cD 
cD = aB 
Make an ABCD diagram : 



AB 


Ab 


aB 


ab 








I 




CD 






1 




Cd 


2 


2 




2 


cD 






1 




cd 



Now, if aB 



Fig. 401. 
cD, then the combinations containing aBO, 



544 



EQUIVALENTS FOR PROPOSITIONS. 



[ Chap. 33. 



aBcd, are inconsistent, and we eliminate them by making a fig- 
ure 1 in those sections. 

Again, if cD = aB, then the combinations containing AcD, 
abcD, are inconsistent, and we eliminate them by making a fig- 
ure 2 in those sections. 

From the consistent combinations we can get the following 
definitions of a letter-term and its negative: 
A = cd | C 
a = bed | bC | BcD 

Make an ABCD diagram : 



AB 


Ab 


aB 


ab 








2 




CD 






2 




Cd 


1 


1 




2 


cD 






2 




cd 



Fig. 402. 

Now, if A = Cd | C, then the combinations containing AcD 
are inconsistent, and we eliminate them by making a figure 1 
in those sections. 

Again, if a = bed | bC | BcD, then the combinations con- 
taining aBcd, aBC, abcD, are inconsistent, and we eliminate 
them by making a figure 2 in those sections. 

The result proves that we have obtained a pair of proposi- 
tions which are equivalent to the pair of given propositions. 

918. Suppose now that we wish to obtain a set of categori- 
cal propositions which shall be equivalent to the pair of prop- 
ositions, 

(1) aB = cD 

(2) cD = aB 



918.] AN EXAMPLE. 

Make an ABCD diagram : 



545 



AB 


Ab 


aB 


ab 








1 

1 




CD 








Cd 


2 


2 




2 


cD 






1 




cd 



Fig. 403. 

Now, if aB = cD, then the combinations containing aBC, 
aBcd, are inconsistent, and we eliminate them by making a fig- 
ure 1 in those sections. 

Again, if cD = aB, then the combinations containing AcD, 
abcD, are inconsistent, and we eliminate them by making a fig- 
ure 2 in those sections. 

From the consistent combinations we can get the following 
categorical definitions : 

(1) Ac = dAc 

(2) BC = ABC 

(3) abc == dabc 

(4) aBc = DaBc 
Make an ABCD diagram : 



AB 


Ab 


aB 


ab 








<> 




CD 






2 




Cd 


1 


1 




3 


cD 






i 




cd 



Fig. 404. 

Now, if Ac = d, then the combinations containing AcD are 
inconsistent, and we eliminate them by making a figure 1 in 
those sections. 



35 



546 



EQUIVALENTS FOR PROPOSITIONS. 



[Chap. 33. 



Again, if BC == A, then the combinations containing aBC 
are inconsistent, and we eliminate them by making a figure 2 
in those sections. 

Again, if abc = d, then the combination abcD is inconsistent, 
and we eliminate it by making a figure 3 in that section. 

Again, if aBc === D, then the combination aBcd is incon- 
sistent, and we eliminate it by making a figure 4 in that section. 

The result proves that we have obtained a set of propositions 
which are equivalent to the given propositions. They are equiv- 
alent to the given propositions for the reason that they cause 
the elimination of exactly the same combinations which the 
given pair of propositions caused us to eliminate, and they 
save exactly the same combinations which the given pair of 
propositions save. 

919. Let the given proposition be, 

A | C = B | D, and conversely. 

Make an ABCD diagram: 



AB 


Ab 


aB 


ab 






4 


2 




CD 


3 






2 


Cd 


1 






4 


cD 




1 


3 




cd 



Fig. 405. 

The proposition A | C = B | D means, 
Ac | aC = Bd | bD. 

Now, if Ac = Bd | bD, then the combinations ABcD, Abed, 
are inconsistent, and we eliminate them by making a figure 1 
in those sections. 

Again, if aC = Bd | Db, then the combinations aBCD, 
abCd are inconsistent, and we eliminate them by making a fig- 
ure 2 in those sections. 

Again, if Bd = Ac ] aC, then the combinations ABCd, aBcd, 



§ 919.] 



AN EXAMPLE. 



547 



are inconsistent, and we eliminate them by making a figure 3 
in those sections. 

Again, if bD — Ac | aC, then the combinations AbCD, abcD, 
are inconsistent, and we eliminate them by making a figure 4 
in those sections. 

The Eeasoning Frame now shows the logical expression of 
the proposition A | C = B | D, and conversely. 

From the consistent combinations which remain we can get 
the following definitions : 

(1) ABC = D 

(2) ABc = d 

(3) AbC = d 

(4) Abe == D 

(5) aBC = d 

(6) aBc == D 

(7) abC = D 

(8) abc = d 

This set of propositions is equivalent to the given proposi- 
tion, 

A | - O = B | D 
Make an ABCD diagram: 



AB 


Ab 


aB 


ab 






3 


5 




CD 


1 

2 






7 


Cd 






8 


cD 




4 


6 




cd 



Fig. 406. 

Now, if ABC == D, then the combination ABCd is incon 
sistent, and we eliminate it by making a figure 1 in that section 

Again, if ABc = d, then the combination ABcD is incon 
sistent, and we eliminate it by making a figure 2 in that section 

Again, if AbC = d, then the combination AbCD is incon 
sistent, and we eliminate it by making a figure 3 in that section 



548 



EQUIVALENTS FOR PROPOSITIONS. 



[ Chap. 33. 



Again, if Abe == D, then the combination Abed is incon- 
sistent, and we eliminate it by making a figure 4 in that section. 

Again, if aBO == d, then the combination aBCD is incon- 
sistent, and we eliminate it by making a figure 5 in that section. 

Again, if aBc = D, then the combination aBcd is incon- 
sistent, and we eliminate it by making a figure 6 in that section. 

Again, if abO = D, then the combination abCd is incon- 
sistent, and we eliminate it by making a figure 7 in that section. 

Again, if abc = d, then the combination abcD is incon- 
sistent, and we eliminate it by making a figure 8 in that section. 

The result proves that this set of eight propositions is equiv- 
alent to the given proposition. 

920. Find the categorical equivalents for this set of propo- 
sitions: 

(1) No CD = a | Ab 

(2) No Cd = A | aB 

(3) No cD = a | AB 

(4) No cd = A | ab. 
Make an ABCD diagram : 



AB 


Ab 


aB 


ab 






1 


1 


1 


CD 


2 


2 


2 
3 




Cd 


3 




3 


cD 


4 


4 




4 


cd 



Fig. 407. 

Now, if No CD = a | Ab, then the combinations containing 
CDa, CDAb, are inconsistent and we eliminate them by making 
a figure 1 in those sections. 

Again, if No Cd = A | aB, then the combinations contain- 
ing ACd, aBCd, are inconsistent, and we eliminate them by 
making a figure 2 in those sections. 

Again, if No cD = a | AB, then the combinations contain- 



§ 920.] 



AN EXAMPLE. 



549 



ing acD, ABcD, are inconsistent and we eliminate them by 
making a figure 3 in those sections. 

Again, if No cd = A | ab, then the combinations contain- 
ing Acd, abed are inconsistent and we eliminate them by mak- 
ing a figure 4 in those sections. 

The Keasoning Frame now shows us that we can get among 
others the following definitions : 
AB = CD 
Ab = cD 
aB == cd 
ab = Cd. 
This set of propositions is equivalent to the given set of 
propositions. 

Make an ABCD diagram: 



AB 


Ab 


aB 


ab 






2 


3 


4 


CD 


1 


2 


3 




Cd 


1 




3 


4 


cD 


1 


2 




4 


cd 



Fig. 408. 

Now, if AB = CD, then the combinations containing ABCd, 
ABc, are inconsistent, and we eliminate them by making a 
figure 1 in those sections. 

Again, if Ab = cD, then the combinations containing AbC, 
Abed, are inconsistent, and we eliminate them by making a 
figure 2 in those sections. 

Again, if aB = cd, then the combinations containing aBC, 
aBcD, are inconsistent, and we eliminate them by making a 
figure 3 in those sections. 

Again, if ab = Cd, then the combinations containing abCD, 
abc, are inconsistent, and we eliminate them by making a 
figure 4 in those sections. 



550 



EQUIVALENTS FOR PROPOSITIONS. 



[Chap. 33. 



The result proves that we have found a set of propositions 
which is equivalent to the given set of propositions. 
921. Let the pair of given propositions be, 

(1) C = A | B 

(2) A = C. 
Make an ABCD diagram: 



•AB 


Ab 


aB 


ab 




1 






1 


CD 


1 






1 


Cd 


2 


' 2 






cD 


2 


2 






cd 



Fig. 409. 

Now, if = A | B, then the combinations containing CAB, 
Cab, are inconsistent, and we eliminate them by making a 
figure 1 in those sections. 

Again, if A == C, then the combinations containing Ac are 
inconsistent, and we eliminate them by making a figure 2 in 
those sections. 

This Reasoning Frame shows that we can get among others 
the following definitions : 

(1) No A = c 

(2) No AB = C 

(3) No ab = C. 
Make an ABCD diagram : 



AB 


Ab 


aB 


ab 




2 






3 


CD 


2 






3 


Cd 


1 


1 






cD 


1 


1 






cd 



Fig. 410. 



§ 922.] 



AN EXAMPLE. 



551 



Now, if No A. === c, then the combinations containing Ac are 
inconsistent and we eliminate them by making a figure 1 in 
those sections. 

Again, if No AB == C, then the combinations containing 
ABC, are inconsistent and we eliminate them by making a 
figure 2 in those sections. 

Again, if No ab = C, then the combinations containing abC, 
are inconsistent, and we eliminate them by making a figure 3 in 
those sections. 

The result proves that we have obtained a triplet of propo- 
sitions which is equivalent to the given pair of propositions. 

922. Find four negative compound propositions, which, 
taken together, shall be equivalent to the following four dis- 
junctive propositions taken together. 

(1) AB = Cd | c 

(2) Cd = A | ab 

(3) Ab = C | cD 

(4) cD = A | aB. 
Make an ABCD diagram: 



AB 


Ab 


aB 


ab 




1 








CD 






2 




Cd 








4 


cD 




3 






cd 



Fig. 411. 

Now, if AB == Cd | c, then the combination ABCD, is incon- 
sistent, and we eliminate it by making a figure 1 in that section. 

Again, if Cd = A | ab, then the combination containing 
aBCd, is inconsistent and we eliminate it by making a figure 2 
in that section. 

Again, if Ab = C | cD, then the combination containing 
Abed is inconsistent and we eliminate it by making a figure 3 
in that section. 



552 



EQUIVALENTS FOR PROPOSITIONS. 



[ Chap. 33. 



Again, if cD = A | aB, then the combination containing 
abcD, is inconsistent and we eliminate it by making a figure 4 
in that section. 

This Seasoning Frame shows that we can get among others 
the following universal negative compound propositions: 

(1) NoAB = CD 

(2) No Ab = cd 

(3) No aB = Cd 

(4) No ab = cD. 

This quartet of propositions is equivalent to the given quar- 
tet of propositions. 

Make an ABCD diagram: 



AB 


Ab 


aB 


ab 




1 








CD 






3 




Cd 








4 


cD 




2 






cd 



Fig. 412. 

Now, if No AB = CD, then the combination ABCD is 
inconsistent and we eliminate it by making a figure 1 in that 
section. 

Again, if No Ab = cd, then the combination Abed is incon- 
sistent and we eliminate it by making a figure 2 in that 
section. 

Again, if No aB = Cd, then the combination aBCd, is incon- 
sistent and we eliminate it by making a figure 3 in that section. 

Again, if No ab = cD, then the combination abcD is incon- 
sistent and we eliminate it by making a figure 4 in that section. 

The result proves that we have found a quartet of universal 
negative compound propositions equivalent to the given quar- 
tet of disjunctive propositions. 



§ 923.] 



AN EXAMPLE. 



553 



923. Find the categorical equivalents for the following 
propositions : 

(1) AB | ab = CD ] cd 

(2) CD | cd = AB | ab. 
Make an ABCD diagram: 



AB 


Ab 


aB 


ab 






3 


3 




CD 


1 






2 


Cd 


1 






2 


cD 




i 


4 




cd 



Fig. 413. 

Now, if AB = CD | ed, then the combinations ABCd, 
ABcD, are inconsistent and we eliminate them by making a 
figure 1 in those sections. 

Again, if ab === CD | cd, then the combinations abCd, abcD, 
are inconsistent and we eliminate them by making a figure 2 in 
those sections. 

Again, if CD = AB | ab,then the combinations AbCD,aBCD, 
are inconsistent and we eliminate them by making a figure 3 in 
those sections. 

Again, if cd = AB | ab, then the combinations Abed, aBcd, 
are inconsistent and we eliminate them by making a figure 4 
in those sections. 

We can now get among others, the following definitions : 
(1) ABC = D 



(2) ABc 


= d 


(3) AbC 


= d 


(4) Abe 


== D 


(5) aBC 


== d 


(6) aBc 


= D 


(7) abC 


= D 


(8) abc 


= d. 



554 



EQUIVALENTS FOR PROPOSITIONS. 



[ Chap. 33. 



These eight propositions taken together are the equivalents 
of the given pair of propositions. 
Make an ABCD diagram: 



AB 


Ab 


aB 


ab 






3 


5 




CD 


1 
2 






7 


Cd 






8 


cD 




4 


6 




cd 



Fig. 414. 

Now, if ABC = D, then the combination ABCd, is incon 
sistent and we eliminate it by making a figure 1 in that section 

Again, if ABc = d, then the combination ABcD, is incon 
sistent and we eliminate it by making a figure 2 in that section 

Again, if AbC = d, then the combination AbCD is incon 
sistent and we eliminate it by making a figure 3 in that section 

Again, if Abe = D, then the combination Abed, is incon 
sistent and we eliminate it by making a figure 4 in that section 

Again, if aBC = d, then the combination aBCD, is incon 
sistent and we eliminate it by making a figure 5 in that section 

Again, if aBc — D, then the combination aBcd is incon 
sistent and we eliminate it by making a figure 6 in that section 

Again, if abC = D, then the combination abCd is incon 
sistent and we eliminate it by making a figure 7 in that section 

Again, if abc = d, then the combination abcD is inconsist 
ent, and we eliminate it by making a figure 8 in that section. 

The result proves that we have obtained an octave of propo 
sitions equal to the given pair of propositions. 

924. Let the given propositions be: 

(1) cd = AB | Ab | aB 

(2) ba = dO | Dc 

(3) No A = C 

(4) aC == bd 

(5) cD = ab. 



§ 924.] 



AN EXAMPLE. 



555 



Find the equivalents. 
Make an ABCD diagram : 



AB 


Ab 


aB 


ab 




3 


3 


4 


4 
2 


CD 


3 


3 


4 




Cd 


5 


5 


5 




cD 








21 


cd 



Fig. 415. 

Now, if cd = AB | Ab | aB, then the combination abed is 
inconsistent and we eliminate it by making a figure 1 in that 
section. 

Again, if ba = dC | Dc, then the combinations abCD, 
abed, are inconsistent and we eliminate them by making a fig- 
ure 2 in those sections. 

Again, if No A = C, then the combinations containing AC 
are inconsistent and we eliminate them by making a figure 3 
in those sections. 

Again, if aC = bd, then the combinations containing aCD, 
aBCd, are inconsistent and we eliminate them by making a fig- 
ure 4 in those sections. 

Again, if cD == ab, then the combinations containing AcD, 
aBcD, are inconsistent and we eliminate them by making a fig- 
ure 5 in those sections. 

We can now get the following definitions: 

(1) No C = D 

(2) abc = D 

(3) Cd — ab 

(4) Dc = ab. 

This quartet of propositions is equivalent to the set of given 
propositions. 



556 EQUIVALENTS FOR PROPOSITIONS. 

Make an ABCD diagram: 



[Chap. 33. 



AB 


Ab 


aB 


ab 




1 


1 


I 


1 


CD 


3 


3 


3 




Cd 


4 


4 


4 




cD 








2 


cd 



Fig. 416. 

Now, if No C = D, then the combinations containing CI) 
are inconsistent and we eliminate them by making a figure 1 in 
those sections. 

Again, if abc = D, then the combination abed is inconsist- 
ent and we eliminate it by making a figure 2 in that section. 

Again, if Cd == ab, then the combinations containing ACd, 
aBCd, are inconsistent and we eliminate them by making a 
figure 3 in those sections. 

Again, if Dc — ab, then the combinations containing AcD, 
aBcD, are inconsistent and we eliminate them by making a fig- 
ure 4 in those sections. 

Both of these sets of propositions are deductions from the 
tenth amendment to the Constitution of the United States. 
The result proves that we have found a set of propositions 
equivalent to the given set of propositions. 

I think we have now given a sufficient number of examples 
to convince the reader that by this method we can find the 
equivalents for any set of complex propositions. 

925. I consider this method for finding the equivalents for 
given propositions to be one of the most important logical dis- 
coveries which I have made. 

By the use of the Reasoning Frame I have discovered 
another easy method of finding equivalent propositions for 
given universal categorical propositions, and propositions of 
the form of Hamilton's Y, i. e., 

Some A is all B. 



j§ 926, 927.] 



AN EXAMPLE. 



557 



926. When the proposition is in the form of All A is some 
B, we change the quality and quantity of B and make all b the 
subject of the new proposition. Then we change the quantity 
and quality of the subject All A, and make Some a, the predi- 
cate. 

Thus, let the proposition be stated : 
A = AB. 
then its equivalent will be, 

b = ba. 

.Make an AB diagram: 



A 


a 








B 


21 




b 



Fig. 417. 

Now, if A = AB, then the combination Ab is inconsistent 
and we eliminate it by making a figure 1 in that section. 

Again, if b = ba, then the combination Ab is inconsistent 
and we eliminate it by making a figure 2 in that section. 

The result proves that the two propositions are equivalent. 

927. Let the given proposition be : 
Some A is all B, 
which can be stated thus : 

BA = B. 

Then we change the quantity and quality of the predicate 
all B into some b, and take it for the subject. 

Then we change the quantity and quality of the subject 
some A into all a, and take it for the predicate. 

The proposition can be stated thus : 
ab = a. 



558 EQUIVALENTS FOR PROPOSITIONS. 

Make an AB diagram : 



[ Chap. 33. 



A 


a 






12 


B 






b 



Fig. 418. 

Now, if B = BA, then the combination Ba is inconsistent, 
and we eliminate it by making a figure 1 in that section. 

Again, if a = ab, then the combination Ba is inconsistent, 
and we eliminate it by making a figure 2 in that section. 
The result proves the equivalence of the two propositions. 
928. Let the given proposition be: 

All A == some BO. 
In this case we shall have to find two propositions which 
shall be together equivalent to the given proposition. The 
given proposition can be stated thus : 
A = ABC. 
Change the quantity and quality of B and of C and make b 
and c the subjects of the new propositions. 

Change the quantity and quality of the subject A and make 
a the predicate of each new proposition. 
The new propositions can be stated thus: 
b = ba 
c == ca. 
Make an ABC diagram: 



AB 


Ab 


aB 


ab 






2 
1 






C 


1 
3 


2 
1 
3 






c 



Fig. 419. 



§ 929.] 



AN EXAMPLE. 



559 



Now, if A = ABC, then the combinations ABc, AbC, Abe, 
are inconsistent and we eliminate them by making a figure 1 in 
those sections. 

Again, if b = ba, then the combinations containing Ab are 
inconsistent and we eliminate them by making a figure 2 in 
those sections. 

Again, if c = ca, then the combinations containing Ac are 
inconsistent and we eliminate them by making a figure 3 in 
those sections. 

The result proves that the pair of propositions which we 
found are equivalent to the given proposition. 

929. Let the given proposition be : 
C = CAB, 

The pair of equivalent propositions will be : 
a = ac 
b = be. 

Make an ABC diagram : 



AB 


Ab 


aB 


ab 






1 
3 


1 
2 


3 

1 


C 










c 



Fig. 420. 

Now, if C = CAB, then the combinations containing CAb, Ca 
are inconsistent and we eliminate them by making a figure 1 
in those sections. 

Again, if a = ac, then the combinations containing aC are 
inconsistent and we eliminate them by making a figure 2 in 
those sections. 

Again, if b = be, then the combinations containing bC are 
inconsistent and we eliminate them by making a figure 3 in 
those sections. 

The result proves that we have found a pair of propositions 
equivalent to the given proposition. 



560 



EQUIVALENTS FOR PROPOSITIONS. 



[ Chap. 33. 



930. Let the given proposition be: 

All AB is some CD. 
It can be stated thus : 

AB = ABCD. 
In this case the subjects of our two new propositions will be 
c and d, and the predicates will be the complete opposite of 
AB, which are cAb, | caB, | cab, and dAb | daB | dab. 
The new propositions can be stated thus : 
c = cAb | caB | cab 
d === dAb | daB | dab. 
Make an ABCD diagram: 



AB 


Ab 


aB 


ab 













CD 


31 






Cd 


21 








cD 


213 








cd 



Fig. 421. 

Now, if AB = ABCD, then the combinations containing 
ABCd, ABc, are inconsistent and we eliminate them by mak- 
ing a figure 1 in those sections. 

Again, if c = CAb | caB | cab, then the combinations con- 
taining ABc are inconsistent and we eliminate them by making 
a figure 2 in those sections. 

Again, if d = dAb | daB | dab, then the combinations 
containing ABd are inconsistent and we eliminate them by 
making a figure 3 in those sections. 

The result proves that the pair of propositions which we 
have found are equivalent to the given proposition. 

931. Let the given proposition be: 
AB = ABCDE. 



§ 931.] AN EXAMPLE. 

The equivalent propositions will be: 

c = cAb | caB | cab 
d == dAb | daB | dab 
e = eiVb | eaB | eab. 

Make an ABODE diagram : 



5G1 



AB 


Ab 


aB 


ab 












CDE 


14 








CDe 


13 








(ME 


4 

13 








Cde 


12 








cDE 


i 
12 








cDe 


3 
12 








cdE 


43 
12 








cde 



Fig. 422. 

Now, if AB = ABODE, then all the combinations contain- 
ing AB, excepting ABODE, are inconsistent and we eliminate 
them by making a figure 1 in those sections. 

Again, if c = cAb | caB | cab, then the combinations con- 
taining ABc, are inconsistent and we eliminate them by mak- 
ing a figure 2 in those sections. 

Again, if d = dAb | daB | dab, then the combinations 
containing ABd are inconsistent and we eliminate them by 
making a figure 3 in those sections. 

Again, if e = eAb | eaB | eab, then the combinations con- 
taining A Be are inconsistent and we eliminate them by mak- 
ing a figure 4 in those sections. 

The result proves that we have found a triplet of proposi- 
tions equivalent to the given proposition. 



36 



CHAPTER XXXIV. 

CONTRADICTORIES OP PROPOSITIONS. 

932. By the use of the Reasoning Frame I have discovered 
an easy method of finding propositions contradictory to a 
given proposition. 

It is as follows: 

First, Obtain the visible expression of the given proposition 
in the Reasoning Frame. 

Second, Observe what combinations it would be necessary 
to eliminate in order to produce the total elimination of a let- 
ter-term. 

Third, Make a similar Reasoning Frame to the one first 
made, and eliminate in it the combinations which were neces 
sary to cause the elimination of a letter-term in the first 
Reasoning Frame. 

Fourth, From the uneliminated combinations in the second 
Reasoning Frame, get definitions which would cause the elim- 
ination of the eliminated combinations. 

Let us take this example: 

A = B | C = D, 
and ascertain by the method above described, what contradic- 
tories to it we can find. 

Make an ABCD diagram: 



AB 


Ab 


aB 


ab 




1 








CD 




1 


1 


1 


Cd 




1 






cD 




1 






cd 



Fig. 423. 



932.] 



AN EXAMPLE. 



563 



Now, if A = B, except where C = D, and C = D, except 
where A = B, then the combinations containing ABOD, 
AbCd, Abe, aCd, are inconsistent and we eliminate them by 
making a figure 1 in those sections. 

The Reasoning Frame now shows the visible expression of 
the proposition : - 

A == B | = D. 

By observation we now learn that if the combinations con- 
taining ABCd, ABc, AbCD were eliminated, then all the A's 
would be eliminated. 

Make an ABCD diagram and eliminate the combinations 
containing ABCd, ABc, AbCD, by making a figure 1 in those 
sections. 

Make an ABCD diagram: 



AB 


Ab 


aB 


ab 






1 






CD 


1 








Cd 


1 








cD 


I 








cd 



Fig. 424. 

From the uneliminated combinations we can get these defi- 
nitions, and they will cause the elimination of the eliminated 
combinations. 

(1) AB = ABCD 

(2) CD = AB | a. 

This pair of propositions is contradictory to the given propo- 
sition, because, with the given proposition it would cause the 
total elimination of the letter A. 

Again, by observation of the Reasoning Frame, No. 423, 
we see that if the combinations containing ABCd, ABc, aBCD, 
aBc, were eliminated, then all the B's would be eliminated. 



564 CONTRADICTORIES OF PROPOSITIONS. 

Make an ABCD diagram: 



[ Chap. 34. 



AB 


Ab 


aB 


ab 








2 




CD 


2 









Cd 
cD 


2 




2 


2 




2 




cd 



Fig. 425. 

Eliminate the combinations containing ABCd, A Be, aBCD, 
aBc, by making a figure 2 in those sections. 

From the uneliminated combinations we can get these defi- 
nitions, and they will cause the elimination of the eliminated 
combinations : 

(1) AB = ABCD 

(2) aB = aBCd. 

This pair of propositions is contradictory to the given propo- 
sition, because it would cause the total elimination of the let- 
ter B. 

Again, by observation of the Reasoning Frame, No. 423, we 
can see that if the combinations containing ABCd, AbCD, aCD 
were eliminated, then all the C's would be eliminated. 

Make an ABCD diagram: 



AB 


Ab 


aB 


ab 






3 


3 


3 


CD 


3 








Cd 











cD 








cd 



Fig. 426. 

Eliminate the combinations containing ABCd, AbCD, aCD, 
by making a figure 3 in those sections. 



§ 933.] A NEW METHOD. 565 

From the uneliminated combinations we can get these defi- 
nitions and they will cause the elimination of the eliminated 
combinations, 

(1) CD == CDAB 

(2) AB = ABCD | ABc. 

This pair of propositions is contradictory to the given propo- 
sition, because it would cause the elimination of the letter C. 

By the same method we can learn that the following pair of 
propositions is contradictory to the given proposition, because 
it will cause the elimination of the letter D, 

(1) AB = ABC | ABcd 

(2) cD = cDAb, 

Also the following proposition is contradictory to the given 
proposition, because it will cause the elimination of the let- 
ter a, 

(1) a = aCd. 

Also the following pair of propositions is contradictory to 
the given proposition, because it will cause the el i mi nation 
of the letter b, 

(1) ab = abCd 

(2) CD = CDB. 

Also the following proposition is contradictory to the given 
proposition, because it will cause the elimination of the letter c, 
(1) c = cAb. 

Also the following pair of propositions is contradictory to 
the given proposition, because it will cause the elimination of 
the letter d, 

(1) cd = cdAb 

(2) AB = ABD. 

933. I have discovered another easy method of finding by 
our system the perfect contradictories of any given proposition. 
The method is as follows : 

(1) Eliminate the inconsistent combinations. 

(2) Make a letter S in the eliminated combinations, for 
the purpose of indicating that those combinations are 
to be read. 



566 



CONTRADICTORIES OF PROPOSITIONS. [ Chap. 34. 



(3) From these combinations which are to be read, get 
definitions according to the method hitherto pursued. 

These definitions thus obtained, will eliminate all the com- 
binations which are consistent with the given proposition, and 
will save all the combinations which are inconsistent with the 
given proposition. Thus, we will have found a set of propo- 
sitions which will be complete contradictories to the given 
propositions. 

934. Find a set of propositions which will be contradictories 
to the following propositions : 

(1) A == AB 

(2) D = DC 
Make an ABOD diagram: 



AB 


Ab 


aB 


ab 






S 
1 






CD 




s 

1 






Cd 


S 
2 


s 
1 

2 


S 
2 


S 
2 


cD 




s 
1 






cd 



Fig. 427. 

Now, if A = B, then the combinations containing Ab are 
inconsistent and we eliminate them by making a figure 1 in 
those sections. 

Again, if D == C, then the combinations containing Dc are 
inconsistent and we eliminate them by making a figure 2 in 
those sections. 

Make a letter S in the eliminated combinations. 

From the eliminated combinations we can get among others, 
these definitions : 

(1) AB = cD 

(2) aB = cD 

(3) ab = cD. 

This set of definitions will eliminate what the given pair of 



§ 935.] 



AN EXAMPLE. 



567 



propositions saved and will save what the given pair of propo- 
sitions eliminated. 

Make an A BCD diagram: 



AB 


Ab 


aB 


ab 




1 




2 


3 


CD 


1 




2 


3 


Cd 










cD 


1 




2 


3 


cd 



Fig. 428. 

Now, if AB = cD, then the combinations containing ABC, 
ABcd, are inconsistent and we eliminate them by making a fig- 
ure 1 in those sections. 

Again, if aB = cD, then the combinations containing aBC, 
aBcd, are inconsistent and we eliminate them by making a fig- 
ure 2 in those sections. 

Again, if ab = cD, then the combinations containing abC, 
abed, are inconsistent and we eliminate them by making a 
figure 3 in those sections. 

The result proves that we have found a set of propositions 
completely contradictory to the given propositions. 

935. Let the given propositions be : 

(1) AB == CD 

(2) CD = AB. 
Make an ABCD diagram : 



AB 


Ab 


aB 


ab 






S 
2 


S 
2 


S 
2 


CD 


S 
1 








Cd 


s 
1 








cD 


s 
1 








cd 



Fig. 429. 



568 



CONTRADICTORIES OF PROPOSITIONS. [ Chap. 34. 



Now, if AB =s CD, then the combinations containing ABo, 
ABCd, are inconsistent and we eliminate them by making a 
figure 1 in those sections. 

Again, if CD = AB, then the combinations containing aCD, 
AbCD, are inconsistent and we eliminate them by making a 
figure 2 in those sections. 

Make a letter S in the eliminated combinations. From the 
eliminated combinations we can get the following definitions : 

(1) Cd = ABCd 

(2) cD = ABcD 

(3) cd = ABcd 

(4) CD = Ab | a. 

This set of propositions is contradictory to the given pair of 
propositions. 

Make an ABCD diagram: 



AB 


Ab 


aB 


ab 




4 








CD 




1 


1 


1 


Cd 




2 


2 


2 


cD 




3 


3 


3 


cd 



Fig. 430. 

Now, if Cd = AB, then the combinations containing aCd, 
AbCd, are inconsistent and we eliminate them by making a 
figure 1 in those sections. 

Again, if cD === AB, then the combinations containing acD, 
AbcD, are inconsistent and we eliminate them by making a 
figure 2 in those sections. 

Again, if cd = AB, then the combinations containing acd, 
Abed, are inconsistent and we eliminate them by making a 
figure 3 in those sections. 

Again, if CD = Ab | a, then the combination ABCD is 
inconsistent and we eliminate it by making a figure 4 in that 
section. 



5§ 936, 937.] 



AN EXAMPLE. 



569 



The result proves that we have found a set of propositions 
completely contradictory to the given propositions. 

936. The propositions which we found in the preceding 
example, are contradictory to the given propositions, but they 
are not the only ones which can be found. We can find several 
different sets of propositions contradictory to the given propo- 
sitions. 

937. Let it be required to find a set of universal affirmative 
propositions which shall be contradictory to the following 
propositions. 

(1) BC = A, or B = | DE 

(2) B = C | DE, or BC = A. 
Make an ABODE diagram: 



AB 


Ab 


aB 


ab 








s 

1 




ODE 


S 
12 








CDe 


S 
21 








CdE 


s 

21 








Cde 


S 
21 








cDE 


S 
2 




s 

2 




cDe 


S 
2 




S 
2 




cdE 


S 
2 




S 
2 




cde 



Fig. 431. 

Now, if BC = A, or B = C | DE, then the combinations 
containing ABCDe, ABCd, ABcDE, aBCDE, are inconsistent 
and we eliminate them by making a figure 1 in those sections. 

Again, if B = C | DE, or BC = A, then the combinations 
containing ABCDe, ABCd, ABc, aBcDe, aBcd, are inconsist- 
ent and we eliminate them by making a figure 2 in those 
sections. 



570 



CONTRADICTORIES OF PROPOSITIONS. [ Chap. 34. 



Make a letter S in the eliminated combinations. 
From the eliminated combinations we can now obtain the 
following definitions: 

(1) CDE = BaCDE 

(2) CDe = ABODe 

(3) CdE == ABCdE 

(4) Cde == ABCde 

(5) cDE = ABcDE 

(6) cDe = BcDe 

(7) cdE = BcdE 

(8) cde s== Bcde. 

This set of propositions will be completely contradictory to 
the given propositions. 

Make an ABODE diagram: 



AB 


Ab 

1 


aB 


ab 




1 




1 


CDE 




2 


2 


2 


CDe 




3 


3 


3 


CdE 




4 


4 


4 


Cde 




5 


5 


5 


cDE 




6 




6 


cDe 




7 




7 


cdE 




8 




8 


cde 



Fig. 432. 

Now, if CDE = Ba, then the combinations containing 
ACDE, abCDE, are inconsistent and we eliminate them by 
making a figure 1 in those sections. 

Again, if CDe = AB, then the combinations containing 
aCDe, AbCDe, are inconsistent and we eliminate them by mak- 
ing a figure 2 in those sections. 



§ 938.] 



AN EXAMPLE. 



571 



Again, if CdE — AB, then the combinations containing 
aCdE, AbCdE, are inconsistent, and we eliminate them by 
making a figure 3 in those sections. 

Again, if Cde = AB, then the combinations containing aCde, 
AbCde, are inconsistent, and we eliminate them by making a 
figure 4 in those sections. 

Again, if cDE — AB, then the combinations containing 
acDE, AbcDE, are inconsistent, and we eliminate them by 
making a figure 5 in those sections. 

Again, if cDe = B, then the combinations containing bcDe 
are inconsistent, and we eliminate them by making a figure 6 
in those sections. 

Again, if cdE = B, then the combinations containing bcdE 
are inconsistent, and we eliminate them by making a figure 7 
in those sections. 

Again, if cde = B, then the combinations containing bcde 
are inconsistent, and we eliminate them by making a figure 8 
in those sections. 

The result proves that we have found an octave of proposi- 
tions completely contradictory to the given pair of proposi- 
tions. 

938. Let it be required to find a set of propositions contra- 
dictory to the following propositions: 

(1) ab = Cd | cD 

(2) Cd | cD = ab 

(3) C = d 
Make an ABCD diagram: 



AB 


Ab 


aB 


ab 




S 
3 


S 
3 


s 

3 


S 
13 


CD 


S 
2 


s 

2 


s 

2 




Cd 


s 

2 


S 


s 

2 




cD 








s 

1 


cd 



Fig. 433. 



572 



CONTRADICTORIES OF PROPOSITIONS. [ Chap. 34. 



Now, if ab = Cd | cD, then the combinations abCD, abed, are 
inconsistent, and we eliminate them by making a figure 1 in 
those sections. 

Again, if Cd | cD = ab, then the combinations containing 
ACd, aBCd, AcD, aBel), are inconsistent, and we eliminate 
them by making a figure 2 in those sections. 

Again, if C == d, then the combinations containing CD are 
inconsistent, and we eliminate them by making a figure 3 in 
those sections. 

Make a letter S in the eliminated sections. 

From the eliminated combinations we can now get the fol- 
lowing definitions: 

(1) A = C | cD 

(2) B — C | cD 

(3) Cd = A | aB 

(4) cD = A | aB 

This set of propositions is completely contradictory to the 
given set of propositions. 
Make an ABCD diagram: 



AB 


Ab 


aB 


ab 












CD 








3 


Cd 








4 


cD 


21 


1 


2 




cd 



Fig. 434. 

Now, if A -- C | cD, then the combinations containing Acd 
are inconsistent, and we eliminate them by making a figure 1 
in those sections. 

Again, if B = C | cD, then the combinations containing Bed 
are inconsistent, and we eliminate them by making a figure 2 
in those sections. 

Again, if Cd = A | aB, then the combination abCd is 



§ 939.] 



AN EXAMPLE. 



573 



inconsistent, and we eliminate it by making a figure 3 in that 
section. 

Again, if cD = A | aB, then the combination abcD is incon- 
sistent, and we elim. ^y making a figure 4 in that sec- 
tion. 

The result now proves that we have found a set of proposi- 
tions completely contradictory to the set of given propositions. 
The given propositions in this case were the propositions rep- 
resenting the tenth amendment to the Constitution of the 
United States. 

939. Let it be required to find a set of propositions which 
shall be contradictory to the following proposition : 

All the combinations are ABCD | ABCd | ABcd | AbCD, 
| AbcD | aBCD | aBcD | aBcd | abCd. 

Make an ABCD diagram : 



AB 


Ab 


aB 


ab 










S 


CD 




S 


S 




Cd 


S 






s 


cD 




s 




s 


cd 



Fig. 435. 

Now, if all the combinations are ABCD | ABCd ' | ABcd 
| AbCD | AbcD | aBCD | aBcD | aBcd | abCd, then the combi- 
nations ABcD, AbCd, Abed, aBCd, abCD, abcD, abed, are 
inconsistent, and we eliminate them by making a letter S in 
those sections. 

Make a letter S in the eliminated sections. 

From the eliminated combinations we can get, among others, 
the following definitions : 

(1) CD = abCD 

(2) AB = cDAB 
. (3) Ab = dAb 



574 



CONTRADICTORIES OF PROPOSITIONS. [ Chap. 34. 



(4) aB = CdaB 

(5) abC = DabC 

This set of propositions is completely contradictory to the 
:iven proposition. 
Make an ABCD diagram : 



AB 


Ab 


aB 


ab 




21 


13 


14 




CD 


2 






5 


Cd 




3 


4 




cD 


2 




4 




cd 



Fig. 436. 

Now, if CD = ab, then the combinations containing ACD, 
a BCD, are inconsistent, and we eliminate them by making a 
figure 1 in those sections. 

Again, if AB = cD, then the combinations containing ABC, 
ABcd, are inconsistent, and we eliminate them by making a 
figure 2 in those .sections. 

Again, if Ab = d, then the combinations containing AbD are 
inconsistent, and we eliminate them by making a figure 3 in 
those sections. 

Again, if aB — Cd, then the combinations containing aBc, 
aBCD, are inconsistent, and we eliminate them by making a 
figure 4 in those sections. 

Again, if abC — D, then the combination abCd is inconsist- 
ent, and we eliminate it by making a figure 5 in that section. 

The result proves that we have found a set of affirmative 
propositions completely contradictory to the given proposi- 
tion. 

940. Let it be required to find a set of propositions com- 
pletely contradictory to the proposition, 

No combinations are BA I Ca I D 



940.] AN EXAMPLE. 

Make an ABCD diagram : 



575 



AB 


Ab 


aB 


ab 






S 






CD 


S 




S 


S 


Cd 




s 


s 


s 


cD 


s 








cd 



Fig. 437. 

Now, if No combinations are BA | Ca | D, then the combina- 
tions containing ABd, AbD, aBCd, aBcD, abCd, abcD, are 
inconsistent, and we eliminate them by making a letter S in 
those sections. 

Make a letter S in the eliminated sections. 

From the eliminated combinations we can get the following 
definitions : 

(1) CD = AbCD 

(2) AB = dAB 

(3) Ab = DAb 

(4) cd = ABcd 

This set of propositions is completely contradictory to the 
given proposition. 

Make an ABCD diagram: 



AB 


Ab 


aB 


ab 




12 




1 


1 


CD 




3 






Cd 


2 








cD 




43 


4 


4 


cd 



Fig. 438. 
Now, if CD = Ab, then the combinations containing aCD, 



576 



CONTRADICTORIES OF PROPOSITIONS. [ Chap. 34. 



ABCD, are inconsistent, and we eliminate them by making a 
figure 1 in those sections. 

Again, if AB == d, then the combinations containing ABD 
are inconsistent, and we eliminate them by making a figure 2 
in those sections. 

Again, if Ab === D, then the combinations containing Abd 
are inconsistent, and we eliminate them by making a figure 3 
in those sections. 

Again, if cd === AB, then the combinations containing acd; 
Abed, are inconsistent, and we eliminate them by making a 
figure 4 in those sections. 

The result proves that we have found a set of propositions 
completely contradictory to the given proposition. 

941. Let it be required to find a set of propositions com- 
pletely contradictory to the following proposition : 
(1) a — BCD 

Make an ABCD diagram : 



AB 


Ab 


aB 


ab 

S 
1 










CD 






S 

1 


s 

1 


Cd 






s 
1 


s 
1 


cD 






s 

1 


s : 
i 


cd 



Fig. 439. 

Now, if a — BCD, then all the combinations containing a, 
excepting aBCD, are inconsistent, and we eliminate them by 
making a figure 1 in. those sections. 

Make a letter S in the eliminated sections. 

From the eliminated combinations we can get the following 
definitions : 

(1) CD = abCD 

(2) Cd = aCd 

(3) C D = acD 

(4) cd = acd 



§ 942.] 



AN EXAMPLE. 



577 



This set of propositions is completely contradictory to the 
given proposition. 
Make an ABCD diagram: 



AB 


Ab 


aB 


ab 




1 


1 


1 




CD 


2 


2 






Cd 


3 


3 






cD 


4 


4 






cd 



Fig. 440. 

Now, if CD = ab, then the combinations containing ACD, 
aBCD, are inconsistent, and we eliminate them by making a 
figure 1 in those sections. 

Again, if Cd = a, then the combinations containing ACd are 
inconsistent, and we eliminate them by making a figure 2 in 
those sections. 

Again, if cD = a, then the combinations containing AcD are 
inconsistent, and we eliminate them by making a figure 3 in 
those sections. 

Again, if cd = a, then the combinations containing Acd are 
inconsistent, and we eliminate them by making a figure 4 in 
those sections. 

In this case, the set of propositions completely contradic- 
tory to the given proposition are inconsistent, because the 
letter A is eliminated. 

942. Let it be required to find a set of propositions contra- 
dictory to the proposition, 

A = Be I bC 



37 



578 CONTRADICTORIES OF PROPOSITIONS. 

Make an ABC diagram : 



[ Chap. 34. 



AB 


Ab 


aB 


ab 




S 
1 








C 




S 

1 






c 



Fig. 441. 

Now, if A = Be | bC, then the combinations ABC, Abe, are 
inconsistent, and we eliminate them by making a figure 1 in 
those sections. 

Make a letter S in the eliminated sections. 

From the eliminated combinations we can get the following 
definitions : 

(1) C = ABC 

(2) c = Abe 

This pair of propositions is completely contradictory to the 
given proposition. 

Make an ABC diagram : 



AB 


Ab 


aB 


ab 






1 


1 


1 


C 


2 




2 


2 


c 



Fig. 442. 

Now, if C = AB, then the combinations containing aC, AbC, 
are inconsistent, and we eliminate them by making a figure 1 
in those sections. 

Again, if c = Ab, then the combinations containing ac, 
ABc, are inconsistent, and we eliminate them by making a fig- 
ure 2 in those sections. 



§ 943.] 



AN EXAMPLE. 



579 



The result proves that we have found a pair of propositions 
completely contradictory to the given proposition. The pair of 
propositions which we have found are contradictories because 
the letter a is eliminated. 

943. Let the given propositions be, 

Ac | aC = Ad | aD 
Ad | aD = Ac | aC 
Let it be required to find, 

(1) A set of propositions contradictory to the given 
propositions. 
Make an ABCD diagram: 



AB 


Ab 


aB 


ab 












CD 


S 
3 


S 
3 


S 
2 


S 
2 


Cd 


S 
1 


S 
1 


S 
4 


S 
4 


cD 










cd 



Fig. 443. 

Now, if Ac = Ad, then the combinations containing AcD are 
inconsistent, and we eliminate them by making a figure 1 in 
those sections. 

The reader will observe that I pay no attention, in this con- 
nection, to the alternant aD. It would be impossible for A to 
equal a, hence, in a case of this kind, the alternant aD is super- 
fluous. 

Again, if aC == aD, then the combinations containing aCd 
are inconsistent, and we eliminate them by making a figure 2 
in those sections. 

Again, if Ad = Ac, then the combinations containing ACd 
are inconsistent, and we eliminate them by making a figure 3 
in those sections. 

Again,if aD = aC, then the combinations containing acD 
are inconsistent, and we eliminate them by making a figure 4 
in those sections. 



580 



CONTRADICTORIES OF PROPOSITIONS. [ Chap. 34. 



The Reasoning Frame now shows the logical expression of 
the combination of the given propositions. 
We can read the results thus : 

(1) c '-= d, d = c 

(2) C = D, D = 

Make a letter S in the eliminated combinations. 
The contradictories are, 

(1) O = d 

(2) c = D 
Make an ABCD diagram : 



AB 


Ab 


aB 


ab 




1 


1 


1 


1 


CD 










Cd 










cD 


1 


1 


1 


1 


cd 



Fig. 444. 

Now, if C = d, then the combinations containing CD are 
inconsistent, and we eliminate them by making a figure 1 in 
those sections. 

Again, if c = D, then the combinations containing cd are 
inconsistent, and we eliminate them by making a figure 2 in 
those sections. 

The result proves that we have found a pair of propositions 
contradictory to the given propositions. 

944. When we have worked an example out in the Reason- 
ing Frame, we can always frame an affirmative proposition 
equivalent to the given propositions, by taking every combi- 
nation for the subject, and all the uneliminated combinations 
in the alternative form for the predicate. 

We can also get a single proposition equivalent to the given 
propositions, which shall have no combination for its subject, 
and all the eliminated combinations in the alternative form for 
the predicate. 



§ 945.] ANOTHER METHOD. 581 

We can also get a single proposition which shall be contra- 
dictory to the given proposition, by framing a proposition 
which shall have every combination for its subject and all the 
eliminated combinations in the alternative form for its predi- 
cate. 

We can also get a single proposition contradictory to the 
given proposition or propositions which shall have no combina- 
tion for its subject and the uneliminated combinations in the 
alternative.f orm for its predicate. 

945. Let us take the tenth amendment to the Constitution 
of the United States for an example: 

It reads, 

"The powers not delegated to the United States by the Con- 
stitution, nor prohibited by it to the states, are reserved to the 
states respectively, or to the people." 

Let a = powers not delegated to the United States, 
b = powers not prohibited to the states, 
C = powers reserved to the states, 
D = powers reserved to the people. 

The propositions can be stated thus : 

(1) ab = Cd | cD 

(2) Cd | cD = ab 

(3) No C = D 

If any one should doubt the second proposition, i. e., Cd | 
cD = ab, it can be easily proved by using the Law of the 
Excluded Middle, for Cd | cD must = ab, or Cd | cD must 
= AB | Ab | aB. 

Now if Cd | cD = AB | Ab | aB, then, since ab = Cd | cD, 
ab === either AB | Ab | aB. But this is impossible by the 
Law of Contradiction, which says that a thing cannot both be 
and not be at the same time, hence it follows that Cd | cD — 
ab. 



582 CONTRADICTORIES OF PROPOSITIONS. 

Make an ABCD diagram : 



[ Chap. 34. 



AB 


Ab 


aB 
3 


ab 




3 


3 


31 


CD 


2 


2 


2 




Cd 


2 


2 


2 




cD 








1 


cd 



Fig. 445. ' 

Now, if ab == Cd | cD, then the combinations abCD, abed, 
are inconsistent, and we eliminate them by making a figure 1 
in those sections. 

Again, if Cd | cD = ab, then the combinations containing 
ACd, aBCd, AcD, aBcD, are inconsistent, and we eliminate 
them by making a figure 2 in those sections. 

Again, if No C == D, then the combinations containing CD 
are inconsistent, and we eliminate them by making a figure 3 
in those sections. 

The Reasoning Frame now shows the logical expression of 
the combination of the given propositions. 

We can now read in the Reasoning Frame the following 
propositions : 

(1) Every combination is ABcd | Abed | aBcd | abCd 
| abcD, which can be translated, 

By the Constitution all powers are either delegated to the 
United states, prohibited to the states, not reserved to the 
states and not reserved to the people, or, 
delegated to the United States, not prohibited to the states, 
not reserved to the states, and not reserved to the people, or, 
not delegated to the United States, prohibited to the states, 
not reserved to the states and not reserved to the people, or, 
not delegated to the United States, not prohibited to the states, 
reserved to the states, and not reserved to the people, or, 
not delegated to the United States, not prohibited to the states, 
not reserved to the states, but reserved to the people. 



§ 945.] AN EXAMPLE. 583 

(2) No combination is, AC | AcD | aBC | aBcD | abCD 
abed, which can be translated thus: 

By the Constitution no power is delegated to the United 
States and reserved to the states, or, 

delegated to the United States and reserved to the people, or, 
prohibited to the states, and reserved to the states, or, 
prohibited to the states and reserved to the people, or, 
reserved to the states and reserved to the people, or, 
not delegated to the United States, not reserved to the states, 
not prohibited to the States and not reserved to the people. 

(3) Every combination is AC | AcD | aBC | aBcD | abCD 
| abed. 

This is contradictory to the given propositions. 

It can be translated thus: 

Every power is delegated to the United States and reserved 
to the States ; or, 

delegated to the United States and reserved to the people, or, 
prohibited to the states and reserved to the states, or, 
prohibited to the states and reserved to the people, or, 
reserved to the states and reserved to the people, or, 
not delegated to the United States, not prohibited to the states, 
not reserved to the states, and not reserved to the people. 

(4) No combination is ABcd | Abed | aBcd | abCd | 
abcD. 

This is contradictory to the given propositions. 

It can be translated thus : 

By the Constitution no power is delegated to the United 
States, not reserved to the States, and not reserved to the peo- 
ple, or, 

not delegated to the United States, prohibited to the states, 
not reserved to the states, and not reserved to the people, or, 
not delegated to the United States, not prohibited to the states, 
reserved to the states, and not reserved to the people, or, 
not delegated to the United States, not prohibited to the states, 
not reserved to the states, and reserved to the people. 

It will be seen that the tenth amendment to the Constitu- 
tion does not recognize the doctrine of concurrent powers. 



584 CONTRADICTORIES OF PROPOSITIONS. [ Chap. 34. 

I think that this method of finding the complete contradic- 
tories of given propositions is an important and useful discov- 
ery. 

946. The reader will now have learned that there is a great 
difference between our system and the old logic. We make an 
exhaustive representation of all the possible propositions which 
can be made out of the terms used in the given propositions 
and then ascertain what propositions are inconsistent to any 
given proposition, and proceed to eliminate the combinations 
representing the inconsistent propositions, by making a figure 
against them in the diagram. 

The first proposition will eliminate a certain number of com- 
binations. These eliminations are complete and final, so far 
as they go. 

The next proposition will eliminate more combinations and 
thus we go on until all the given propositions have had their 
say and the result shows the survivors. 

947. Eulerian circles will answer for the simple cases used 
in syllogisms, but they are quite useless when we come to work 
with complex propositions involving five or six terms. 

While experimenting with Eulerian circles, in solving logical 
problems, I discovered the method of squares. At that time I 
had no idea that any one else had ever thought of squares as 
a means of solving logical problems. 

948. Every logician has recognized the fact that an affirma- 
tive proposition can be put into a negative form, but in our sys- 
tem, the negative terms are on a par with affirmative ones, and 
are as uniformly developed and used as are affirmative ones. 

Affirmative terms, with us, have no special privileges. In 
working our examples we ask one question, What combina- 
tions are inconsistent with the given proposition? 

949. The common logic talks of "the conclusion," as if there 
were but one conclusion. 

Our system shows that there are many conclusions. 

Our conclusions are so many various modes of expression. 
The same conclusion substantially, is expressed in a great 
many different forms. 



§§ 950-953.] ANOTHER METHOD. 585 

950. The old system is limited to three terms. 

It makes little difference to us how many terms there are. 
The more propositions we have on a given subject, the more 
able we are to get rid of all ambiguities and to make explicit in 
language everything that is implied in the thought. 

951. Our rules, like the rules of practical arithmetic, do 
away with the tediousness and uncertainty of mental calcu- 
lations made with great labor and by which different persons 
arrive at contradictory results without any one of them being 
able to show how the others have made a mistake. 

952. Our system cuts of all debate and brings the parties at 
once to either admit or deny our fundamental principles. 

953. I also frequently use the following method to get the 
complete contradictory of a very complex proposition. 

First, make the proper diagram on common paper and elimi- 
nate the inconsistent combinations so as to get the visible 
expression of the given proposition. 

Second, make a similar diagram on a piece of tracing paper. 
Place the diagram on the tracing paper, over the diagram on 
the common paper, and in the diagram on the tracing paper 
make a figure 1 in the combinations which are uneliminated 
in the diagram on the common paper. We now have in the 
diagram on the tracing paper, the visible expression of the 
complete contradictory of the given proposition. 

By reading the uneliminated combinations in the contradic- 
tory of the given proposition, we can get all the propositions 
which are inconsistent with the given proposition. 



CHAPTER XXXV. 

FALLACIES. 

954. A fallacy is a false or inconclusive reasoning. There 
are two kinds of fallacies. A fallacy is termed formal, when 
it is in the form of expression. When the proposition is not 
true, it is called a material fallacy. Logic, really, has nothing 
to do with material fallacies. 

955. Examples of fallacies are: 

(1) Money is wealth, 

Corn is not money, therefore, 
Corn is not wealth. 

(2) Every tree is a vegetable, 
Grass is not a tree, therefore, 
Grass is not a vegetable. 

(3) Horses are animals, 

Sheep are not horses, therefore, 
Sheep are not animals. 
As these examples are similar we will simply work out the 
first one: 
Let A = money, 
B — wealth, 
C = corn. 
The propositions can be stated thus: 

(1) A = AB 

(2) C = Ca 



956.] EQUIVOCATION. 

Make an ABC diagram: 



587 



AB 


Ab 


aB 


ab 




2 


21 






C 




1 






c 



Fig. 446. 

Now, if A = AB, then the combinations containing Ab are 
inconsistent, and we eliminate them by making a figure 1 in 
those sections. 

Again, if C = Ca, then the combinations containing CA are 
inconsistent, and we eliminate them by making a figure 2 in 
those sections. 

The Keasoning Frame now shows that the definition of C is : 

= aB | ab, which can be translated, 
Corn is not money and it is wealth or not wealth. 
956. The fallacy of equivocation is where the middle term 
is used in two different senses. An example is, 
Repentance is a good thing. 
Wicked men abound in repentance, therefore, 
They abound in what is good. 
In the first proposition repentence means genuine sorrow ; in 
the second it means regret arising from pain or loss. 
The premises should be stated as follows: 
A = AB 
C = CD 



588 



FALLACIES. 



[ Chap. 35. 



Make an A BCD diagram: 



AB 


Ab 


aB 


ab 






1 






CD 


2 


21 


2 


2 


Cd 




1 






cD 




1 






cd 



Fig. 447. 

Now, if A = AB, then the combinations containing Ab are 
inconsistent, and we eliminate them by making a figure 1 in 
those sections. 

Again, if = CD, then the combinations containing Cd are 
inconsistent, and we eliminate them by making a figure 2 in 
those sections. 

The Reasoning Frame now shows that the definition of C is, 
C === ABD | aBD | abD, 
which can be translated, 

Wicked men abound in regret, and they do or do not 
abound in genuine sorrow, and they do or do not 
abound in a good thing. 

957. The fallacy of reasoning in a circle is where a person 
pretends to prove the truth of a proposition by asserting the 
truth of the conclusion. It is like saying A is B because it 
isB. 

Whately gives this example: 

"To allow every man an unbounded freedom of speech must 
always be, on the whole, advantageous to the state; for it is 
highly conducive to the interests of the community that each 
individual should enjoy liberty, perfectly unlimited, of express- 
ing his sentiments." 

This is rant, not reasoning. 

958. The fallacy of Petitio Princippii, or begging the ques- 



§§ 959-962.] SELF CONTRADICTION. 589 

tion, is where a person reasons on a supposition which is not 
proved or granted. 

959. The fallacy of self-contradiction is where arguments 
are advanced which contradict themselves. 

An example is: "There are three points in this case," said 
the defendant's counsel ; "in the first place we contend that the 
kettle was cracked when we borrowed it; secondly, that it was 
whole when we returned it; and, thirdly, that we never had it 
at all." 

960. The fallacy of Ignoratio ElencM, or irrelevant conclu- 
sion, is when a conclusion is substituted for the one which 
ought to have been proved. An example is : 

The fine arts please the imagination and adorn and pol- 
ish life. 

But the fine arts are the parents of luxury, therefore, 
the fine arts are a frivolous amusement. 

961. The fallacy of the Argumentnm ad Hominem is where a 
reference is made to something in the condition of the person 
who is addressed, to prove the truth of the argument. This 
argument is fair when it is applied solely to the principles of 
the person spoken to. Christ once used it with telling effect 
on the Pharisees. Luke's gospel, chap. 13, v. 5. The Phari- 
sees pretended to be scandalized because Christ did works of 
mercy on the Sabbath. He said to them: "Which of yon 
shall have an ass or an ox fallen into a pit, and will not 
straightway pull him out on the Sabbath day?" 

962. The fallacy of Confusion of Ideas is where a person 
gets mixed up and perplexed in his reasoning. 

A tricky man went into the shop of a rather simple-minded 
woman and asked for a penny loaf and a penny glass of gin. 
The articles being given, he drank the gin and addressed the 
woman as follows: 

"On second thoughts, I will not take the bread ; therefore, I 
just give it back in payment of the gin." The woman, some- 
what perplexed, answered: "But you did not pay me for the 
bread." "Well," said the man, "I have not taken it." "But 



590 FALLACIES. [ Chap. 35. 

where is the payment for the gin?" "My good woman," 
replied the man, "haven't I told you already that I have given 
you back the penny loaf for it?" 

This piece of sophistry so confused the ideas of the poor 
woman that she allowed the villain to depart. 

A herring and a half for three half pence, how many for 
eleven pence? has perplexed many people at first sight. 

963. The fallacy of Suppressio Veri, or the suppression of 
truth, is a common and dishonest way of reasoning. An 
example is, where a person was openly accused in an assembly 
of being concerned in appropriating the public money. 
Another rose to refute the calumny, and said that the accused 
was a most estimable individual. He was a good father and 
an exemplary husband, but not one word on the actual merits 
of the question, and by this sort of clap-trap appeal he pre- 
vented all inquiry as to the charge made. 

The foregoing examples have been selected from Chambers' 
"Information for the People." 

964. Let us take this example of a fallacy: 

(1) You are not what I am 

(2) I am a man, therefore, 
You are not a man. 

Let A = you 

B = I 

C == man 
The premises can be stated thus : 

(1) A = Ab 

(2) B = BC 



§ 965.] AN EXAMPLE. 

Make an ABC diagram : 



591 



AB 


Ab 


aB 


ab 




1 








C 


12 




2 




c 



Fig. 448. 

Now, if A = Ab, then the combinations containing AB are 
inconsistent, and we eliminate them by making a figure 1 in 
those sections. 

Again, if B = BC, then the combinations containing Be are 
inconsistent, and we eliminate them by making a figure 2 in 
those sections. 

The Reasoning Frame now shows that the definition of A is, 
A = bC | be, 
which can be translated thus : 

You are a man or not a man. 
Let us take this example of a fallacy: 
(1) Italy is a Catholic country and abounds in beggars, 
France is also a Catholic country; therefore, 
France abounds in beggars. 
A = Italy 

B = Catholic country 
C = abounds in beggars 
D = France 
The premises can be stated as follows: 

(1) A = ABC 

(2) D = DB 

(3) No A = D 
I assume that Italy and France are different countries. 



965. 



Let 



592 FALLACIES. 

Make an ABCD diagram : 



[Chap. 35. 



AB 


Ab 


aB 


ab 




3 


312 




2 


CD 




1 






Cd 


31 


312 




2 


cD 


1 


1 






cd 



Fig. 449. 

Now, if A = ABC, then the combinations containing ABc, 
Ab, are inconsistent, and we eliminate them by making a fig- 
ure 1 in those sections. 

Again, if D = DB, then the combinations containing Db are 
inconsistent, and we eliminate them by making a figure 2 in 
those sections. 

Again, if No A = D, then the combinations containing AD 
are inconsistent, and we eliminate them by making a figure 3 
in those sections. 

The Seasoning Frame now shows that the definition of D is, 
D =aBC | aBc, 
which can be translated, 

France is not Italy and it is a Catholic country, and it 
abounds or it does not abound in beggars. 
These examples demonstrate that if we know how to state 
propositions correctly, our system will always enable us to 
detect fallacies. 

966. Let us take this example: 

Two and three are even and odd, 
Five are two and three, therefore, 
Five are even and odd. 
These premises really mean, 

(1) Two and three taken separately are even and odd. 

(2) Five is two and three taken together. 



EXAMPLES. 



593 



Let A = two and three taken separately, 

B = even and odd, 

C = five, 

D = two and three taken together. 
The premises may be stated thus : 

(1) A = AB 

(2) C = CD 
Make an ABCD diagram: 



AB 


Ab 


aB 


ab 






1 






CD 


2 


21 


2 


2 


Cd 




1 






cD 




1 






cd 



Fig. 450. 

Now, if A = AB, then the combinations containing Ab are 
inconsistent, and we eliminate them by making a figure 1 in 
those sections. 

Again, if C = CD, then the combinations containing Cd are 
inconsistent, and we eliminate them by making a figure 2 in 
those sections. 

The Reasoning Frame now shows us that the definition of 
Cis, 

C = D (A | a) (B | b) 
which can be translated, 

Five is equal to two and three taken together 
967. Let us take this example: 

All the musical instruments of the Jewish temple made 
a noble concert. 
The harp was a musical instrument of the Jewish temple, 
therefore, 

The harp made a noble concert. 
In the first premise the word "all" is collective, and not uni- 

38 



594 FALLACIES. [ Chap. 35. 

versal. It does not mean each and every. The subject of the 
premise is, 

All the musical instruments of the Jewish temple. 
Let A = all the musical instruments, etc., 
B = a noble concert, 
C == harp, 

D = a musical instrument, etc. 
The premises can be stated thus: 

(1) A == AB 

(2) C == CD 

The symbolic conclusion in this case will be similar to the 
one in the preceding case, 

C = D (A | a) (B | b) 
and it can be translated thus : 

The harp was a musical instrument of the Jewish 
temple, and made or did not make a noble concert. 

968. Let us take this example: 

All animals were in Noah's Ark, therefore, 

No animals perished in the flood. 
In this case the word "all** means every kind of, and refers 
to species. The word "no" refers to individual animals, hence, 
it is plain that the conclusion given is not warranted by the 
premise. 

969. Let us take this example: 

He that sends forth a book into the light, desires it to be 

read. 
He that throws a book into the fire, sends it into the 

light, therefore, 
He that throws a book into the fire desires it to be read. 
(Watts' Logic, p. 322.) 
In this case the word "light" in the major proposition means 
the public view of the world. In the minor proposition it 
signifies the brightness of the fire. 
Let A = he that sends forth, etc., 
B = desires it to be read, 



§§ 970-972.] BAIN ON RELATIVITY. 595 

C = lie that throws a book, etc., 
D == sends it into the light. 
The propositions can be stated thus: 

(1) A == AB 

(2) C == CD 

When this is worked out in the Reasoning Frame, we can 
then read, 

He that throws a book into the fire sends it into the 
light, and desires or does not desire that it be read. 

970. A similar example is, 

He who thrusts a knife into another person should be 
punished. 
A surgeon in operating does so, therefore, 
He should be punished. 
The major premise means, 

He who thrusts a knife into another person maliciously. 
The minor premise means, 

A surgeon in operating does so without malice. 
Hence, there are four terms, and the premises can be stated, 
A = AB 
C = CD 
and then we can draw this conclusion, therefore, 
He should be or should not be punished. 

971. Dr. Bain in his work on Logic, makes some interest- 
ing remarks on fallacies. 

He says, "A large class of fallacies consists in denying or 
suppressing the correlatives of an admitted fact. According 
to Relativity, the simplest affirmation has two sides; while com- 
plicated operations may involve unobvious correlates. Thus, 
the daily rotation of the starry sphere is either a real motion of 
the stars, the earth being at rest, or an apparent motion caused 
by the earth's rotation. Plato seems to have fallen into the con- 
fusion of supposing that both stars and earth moved concur- 
rently, which would have the effect of making the stars, to 
appearance, stationary." 

972. "Every mode of stating the doctrine of innate ideas, 



596 FALLACIES. [ Chap. 35. 

commits or borders upon a Fallacy of Relativity, provided we 
accept the theory of Nominalism. A general notion is the 
affirmation of likeness among particular notions ; it, therefore, 
subsists only in the particulars. It cannot precede them in the 
evolution of the mind ; it cannot arise from a source apart, and 
then come into their embrace. A generality not embodied in 
particulars is a self-contradiction, unless on some form of 
Realism." 

973. Kant's autonomy, or self-government of the will, is 
a fallacy of suppressed relative. No man is a law to himself; 
a law co-implicates a superior who gives the law and an inferior 
who obeys it; but the same person cannot both be ruler and 
subject in the same department." 

974. "A fallacy of Relativity is pointed out, by Mr. Venn, 
in the doctrine of Fatalism; a doctrine implying that events, 
depending upon human agency, will yet be equally brought 
to pass, whether men try to oppose or try to forward them." 

(Logic of Chance, p. 366.) 

975. "Fallacies of Relativity often arise in the hyperboles of 
Rhetoric. In order to reconcile to their lot the more humble 
class of manual laborers, the rhetorician proclaims the dignity 
of all labor, without being conscious that if all labor is digni- 
fied, none is; dignity supposes inferior grades, a mountain 
height is abolished if all the surrounding plains are raised to 
the level of its highest peak. 

So, in spurring men to industry and perseverance, examples 
of distinguished success are held up for universal imitation 
while, in fact, these cases owe their distinction to the general 
backwardness." 

(Bain's Logic, pp. 621-22.) 

976. The fallacy of the Irrelevant Question occurs when a 
person is decoyed into committing himself to a categorical 
answer — "Have you cast your horns?" If you answer, "I 
have," it is rejoined, "Then you have had horns;" if you 
answer, "I have not," it is rejoined, "Then you have them 
still?" 



§§ 977-979.] EXAMPLES. 597 

977. Another sophism is the fallacy of putting more ques- 
tions than one as one. 

An example is, 

"Why did you strike your father?" 

978. Prof. DeMorgan, in his work on Formal Logic, has an 
excellent chapter on Fallacies. He gives on page 242, the fol- 
lowing examples of ambiguities : 

Every dog runs on four legs, 

Sirius (the dog star) is a dog, therefore, 

Sirius runs on four legs. 

Nothing is better than wisdom and virtue, 

Dry bread is better than nothing, therefore, 

Dry bread is better than wisdom and virtue. 

A mouse eats cheese, 

A mouse is one syllable, therefore, 

One syllable eats cheese. 

In these examples there are four terms. They can be stated 

thus: . 

A = AB 

C = CD 

and when worked out the conclusions will be, 

Sirius runs or does not run on four legs. 

Dry bread is better or not better than wisdom and virtue. 

One syllable eats or does not eat cheese. 

979. De Morgan gives this as the most difficult example: 
To call you an animal is to speak truth, 

To call you an ass is to call you an animal, therefore, 

To call you an ass is to speak truth. 
I think that these propositions mean, 

You are an animal, 
# An ass is an animal, therefore, 

You are an ass. 
Let A = you, 

B = animal, 

O = ass. 
The propositions can be stated thus : 

A = AB 
C = CB 



598 FALLACIES. 

Make an ABO diagram : 



[ Chap. 35. 



AB 


Ab 


aB 


ab 






12 




2 


C 




1 






c 



Fig. 451. 

Now, if A = AB, then the combinations containing Ab are 
inconsistent, and we eliminate them by making a figure 1 in 
those sections. 

Again, if = CB, then the combinations containing Cb are 
inconsistent, and we eliminate them by making a figure 2 in 
those sections. 

The Reasoning Frame now shows that the definition of A is: 
A = BO | Be, 
which can be translated, 

You are or are not an ass. 

980. DeMorgan says, on page 248, "It must be remembered 
that the word "all" in a proposition is not necessarily signifi- 
cative of a universal proposition; it may be a part of the 
description of the subject, thus, in 

"All the peers are a House of Parliament," 
we do not use the words, "All the peers" in the same sense as 
when we say "All the peers derive their titles from the crown. " 

In the second case the subject of the proposition is "peer" 
and the term "all" is distributed, synonymous with each and 
every. 

In the first case the subject is, "All the peers," and the term 
"all" is collective. 

981. What you bought yesterday, you eat today. You 
bought raw meat yesterday, therefore, you eat raw meat today. 

Let A = you, 
B = bought, 
C = what, 



§981.1 



AN EXAMPLE. 



599 



D = yesterday, 
4£ = eat, 
F = today, 
G = raw meat. 
The premises can be stated thus : 

(1) CABD = CAEF 

(2) A = BDG 
Make an ABCDEFG diagram: 



ABC 


ABc 


AbC 


Abe 


aBC 


aBc 


abC 


abc 








2 


2 










DEFG 


2 


2 


2 


2 










DEFg 


1 




2 


2 










DEfG 


21 


2 


2 


2 










DEfg 


1 




2 


2 










DeFG 


21 


2 


2 


2 










DeFg 


1 




2 


2 










DefG 


21 


2 


2 


2 










Defg 


2 


2 


2 


2 










dEFG 


2 


2 


2 


2 










dEFg 


2 


2 


2 


2 










dEfG 


2 


2 


2 


2 










dEfg 


2 


2 


2 


2 










deFG 


2 


2 


2 


2 










deFg 


2 


2 


2 


2 










defG 


2 


2 


2 


2 










defg 



Fig. 452. 



600 FALLACIES. [Chap. 35. 

Now, if CABD = CAEF, then the combinations containing 
ABCDEf, ABCDe, are inconsistent, and we eliminate them by 
making a figure 1 in those sections. 

Again, if A = BDG, then all the combinations containing A 
excepting those containing ABDG, are inconsistent, and we 
eliminate them by making a figure 2 in those sections. 

The Reasoning Frame now shows that the definition of A is : 

A = BCDEFG | BcDEFG | BcDEfG | BcDeFG | BcDefG 
which can be translated, 

You eat or do not eat today or not today, raw meat. 

982. Another fallacy may be called the fallacy of the Imper- 
fect Dilemma. 

DeMorgan gives this example: 

A body must either be in the state A, or in the state B, 
It cannot change in the state A, it cannot change in the 

state B, therefore, 
It cannot change at all. 
Now, if the alternative A or B be necessary, the correct state- 
ment may be, 

A body must be either in the state A, or in the state B, 
or in the state of transition from one to the other. 

983. Of this kind is the celebrated sophism of Diodorous 
Cronus, that motion is impossible, for all that a body does, it 
does either in the place in which it is, or in the place in which it 
is not and it cannot move in the place in which it is and certainly 
not in the place in which it is not." 

This is an imperfect dilemma, because motion means transi- 
tion from the place in which a body is, to the place in which it 
is not now but will be. 



CHAPTER XXXVI. 

UTILITY OF LOGIC. 

984. The extravagant claims which were sometimes put 
forward bj the advocates of the old logic, often ended in disap- 
pointment and brought the study of logic into disrepute. They 
represented logic as furnishing the only means for the dis- 
covery of all kinds of truth. Of course these pretensions were 
unfounded and the result was, that for a time the study of 
logic was generally abandoned. 

985. I have already said that logic will not discover new 
facts, but our system will furnish a great many new meanings 
of old facts, and often this will be just as important to man- 
kind as the discovery of a new fact. 

986. Many good people think that in order to reason cor- 
rectly it is only necessary to have common sense. Now, while 
common sense is an excellent quality and very necessary to a 
logician, it will not furnish him with those rules which are 
necessary to make a good logician. 

987. It seems to me that the power to reason is the crowning 
glory of man and that it is of the utmost importance that he 
should learn to reason correctly. Every branch of learning 
has its necessary rules, — law, medicine, architecture, engineer- 
ing, navigation, etc., etc., and a man might just as well expect 
to be a good lawyer, doctor, architect, engineer, navigator, 
etc., etc., without a knowledge of the technical rules of those 
subjects, as to be a good reasoner without learning the rules of 
logic, and just as the rules in those branches of science will 
tend to make a man a proficient and useful member of society 
in those professions, so will logic increase his usefulness in 
whatever profession he may engage. 

988. The ability to reason correctly lies at the foundation 
of a knowledge of all the sciences, and the easier it is for a man 
to reason well, the easier will it be for him to master any branch 



602 UTILITY OF LOGIC. [Chap. 36. 

of learning. I believe that any ordinarily intelligent person 
can by our system become a first-class reasoner. If any one 
will master it thoroughly, he will be able to reason as correctly 
as the greatest logician who has ever lived. 

989. This system will also afford a great deal of pleasure to 
those who learn it, by enabling them to combine different prop- 
ositions and from the results of their combination get many 
new and hitherto unsuspected truths. 

990. But its principal utility will lie in the fact that it will 
save a great deal of time and labor in bringing men who admit 
the premises to unanimous conclusions. 

Where the principles are agreed on and the facts are admit- 
ted, it ought to put an end to all dispute. When it becomes 
generally known, I believe that it will be the means of bringing 
men to a substantial agreement on nearly all the disputed 
questions in law, theology, political economy, ethics and kin- 
dred sciences. Its usefulness to disputants and polemic wri- 
ters is indisputable. 



CHAPTER XXXVII. 



PROBLEMS. 



991. 

sions to 



992. 

993. 

994. 

995. 

996. 

997. 

998. 

999. 

1000. 

1001. 

1002. 

1003. 

1004. 

1005. 

1006. 

1007. 



The reader is expected to find the categorical conclu 
the following problems: 



All A is some B 

All B is all C. 

No A is B 

All B is some C. 

No A is B 

No B is C. 

Some A is all B 

All B is some C. 

All A is some not-B 

Some not-B is all C. 

All A is some B 

All not-C is some not-A. 

All not-A is some not-B 

All not-B is some not-C. 

All A is some B 

All C is some A. 

All A is some not-B 

All C is some B. 

All not-A is some B 

All C is some B. 

All not-A is some B 

All not-B is some C. 

All A is some not-B 

All not-B is some C. 

All not-A is some not-C 

All not-B is some C. 

All A is all B 

All B is all A 

All C is some D. 

All AB is some C 

All Ab is some D 

All a is some BCD. 

All A is some BCD 

All a is some b 

All c is some D. 

All a is some BC 

All C is some D 



604 



PROBLEMS. 



[ Chap. 37. 



1008. 
1009. 
1010. 

1011. 

1012. 

1013. 



1014. 



1015. 



1016. 

1017. 
101S. 
1019. 
1020. 



All D is some A. 

All a is some C 

All b is some D 

All D is some c. 

All a is some bed 

All A is some O 

All Ab is some Cd. 

'All AB is some CD 

All Ab is s6me cd 

All aB is some Cd 

All ab is some cD 

All ABC is some D 

All BCD is some e 

All a is some bede 

All Ab is some c. 

All A is some b 

All a is some B 

All D is some f 

All dE is some F 

All bC is some def. 

All ABCD is some ef 

All ABCd is some ef 

No A is b 

No a is B 

No D is ef 

No b is EF 

All a is all d 

All Be is some D. 

If A is AB, then A = C 

If C is CA, then C = B 

If A is AC, then B is C. 

If A is Ab, then A is c 

If c is ca, then c is B 

If a is aB, then B is c 

If a is aC, then a is b 

If A is AB, then A is C 

If C is Ca, then C is b. 

A or not-B is C 

c is Ab. 

A or BC is D. 

C is a or b. 

ABC or aBc is D. 

A or B or C is D. 

A or not-B or not-C is D 

A is BC or D 

AB or C is D. 



1021. AisB orCisD. 



;§ 1022-1044.] 


. PROBLEMS. 


605 


1022. 


is B or A is not-D. 




1023. 


No 


A is Cd or c. 




1024. 


No 


A or B is C or D. 




1025. 


No 


A or B is CD or cd. 




1026. 


No 


AB or ab is C or D. 




1027. 


No A is B or C is D. 




1028. 


C is B or A is not-B. 




1029. 


A, 


B, or C is D. . 




1030. 


A or B or C is D or E. 




1031. 


Ai 


ls B or C is D. 




1032. 


B is C or A is not-D. 




1033. 


A< 


h* B is no C or D. 




1034. 


(1) 


A or B is B or C. 






(2) 


B or C is A or B. 




1035. 


(1) 


A or B is not-b or C. 






(2) 


b or C is A or B. 




1036. 


(1) 


A or B is A or C. 






(2) 


A or C is A or B. 




1037. 


(1) 


AB or CD is E 






-(2) 


E is AB or CD. 




1038. 


(1) 


A is B or B is C 






(2) 


B is C or A is B. 




1039. 


(1) 


If A is B, E is F 






(2) 


If C is D, E is F 






(3) 


But A is B and C is D. 




1040. 


(1) 


A or B is C 






(2) 


C is A or B 


* 




(3) 


A or B is D 






(4) 


D is A or B. 




1041. 


(1) 


A or not-C is B or not-D 






(2) 


B or not-D is A or not-C. 




1042. 


(1) 


A is not-D or B is D 






(2) 


B is D or A is not-D 






(3) 


C is not-D or A is B 






(4) 


A is B or C is not-D. 




1043. 


(1) 


AB is ABC or ABD or ABE 






(2) 


No A is BC or BD or BE. 




1044. 


Give the logical expression in the Reasoning 
of the following propositions: 


Frame 






Some AB is all not-C 








Some A is all not-B 








Some A is not either B or C 








All A is all B 








Some A is all B 








Some A is not any B 








Not any A is B 








Not some A is all B. 





606 PROBLEMS. [ Chap. 37. 

1045. Give the categorical premises which will produce the 
following conclusions: 

ABc | Abe | a Be | abC. 

1046. What categorical premises will produce the following 
conclusion? 

ABcD | AbCd | aBCD | aBcd | abcD 

1047. What categorical premises will produce the follow- 
ing conclusion: 

Everything is ABCde | ABcde | AbCde | Abcde | aBCde 
| aBcde | abCde | abcDE. 

1048. The fact A is always to be found in company with the 
fact C and the fact D, but never in company with the fact B. 

The facts C and D never occur together except in company 
with the fact A, what can we infer about the fact C? 

1049. Wherever there is the circumstance A, there is also the 
circumstance B or the circumstance C. Wherever the circum- 
stances A and C are both present, the circumstance B will also 
be present. The circumstance B never occurs without either 
the circumstance A or the circumstance O. What can you 
infer from the circumstance A, from circumstances where 
there is no A, from circumstances where there is no C? 

1050. If the facts A and B are always found together, and 
if the facts D and E are never found together, and if the 
absence of A makes the presence of B, what relation is there 
between E and B? 

1051. If A is B, A is CDe or CdE, 

If A is bE, A is neither D nor C, 
Ad is c 
Ac is d, 
what can we learn about B and E? 

1052. (1) A is AB 

(2) bA is bAC 

(3) CA is CAD 

(4) No EA is D or a 
What can we learn about D and a? 

1053. Four hunters went on a hunting trip. They were in 
camp seven days. A different party went hunting each day. 
B and D always went out together or stayed in together. 
Name the different parties that went out on seven different 
days. 



APPENDIX. 

HISTORICAL NOTES. 

1054. Some of my readers may not have access to a library 
containing the principal works on logic, and for their benefit, I 
have compiled the following brief historical notes on logic 
and logicians, and a few extracts from works on logic. 

For the first scientific treatment of logic, we are to look to 
the Greeks. Zeno of Elea is called the father of logic and 
dialectics, but it was then treated with particular reference to 
the art of disputation, and soon degenerated into sophistry. 
The Sophists and the Megarean school (founded by Euclid of 
Megara), greatly developed this art. The latter, therefore, 
became known under the name of the heuristic or dialectic 
school, and is famous for the invention of several sophisms." 
(Encyclopedia Americana, vol. 8, p. 49.) 

1055. "The first attempt to represent the forms of thinking, 
in abstracto, on a wide scale and in a purely scientific manner, 
was made by Aristotle. His logical writings were called, by 
later ages, organon, and for almost two thousand years after 
him, maintained authority in the schools of the philosophers. 
His investigations were directed, at the same time, to the cri- 
terion of truth, in which path Epicurus, Zeno, the founder of 
the Stoic school, Chrvsippus, and others, followed him." 

(Ib.) 

1056. "With Epicurus, words were the signs of things, and 
not, as with the Stoics, of the ideas of things. One sought for 
the meanings of words, the other for a knowledge of things." 

(Studies in Logic, p. 1.) 

1057. "Perhaps the most interesting of these early thinkers, 
so far as the history of logic is concerned, is Antisthenes, 
whose extreme nominalism presents the most curious analo- 
gies to some recent logical work. According to Antisthenes * 
* * there is. therefore, no distinction of subject and predi- 
cate possible, even identical propositions, the only possible 
forms under this theory, are mere repetitions of the complex 
name. Predication is either impossible or reduces itself to 
naming in the predicate what is named in the subject. It is 
the simple result of so consistent a nominalism that all truth is 
arbitrary or relative; there is no possibility of contradiction, 
not even of one's self." 

(Encyclopedia Britannica, Vol. 14, p. 786.) 



608 APPENDIX. [ §§ 1058-1062. 

1058. "In essence, the Stoic doctrine is identical with that 
of Antisthenes above noted, and it is interesting to observe 
that, under the purely nominalistic theory, logic became almost 
identical with the doctrine of expression or rhetoric." 

(lb. p. 791.) 

1059. "Aristotle, one of the most celebrated philosophers of 
Greece, and the founder of the Peripatetic sect, was born at 
Stagyra, a town of Thrace, B. C. 384, being the son of Nicoma- 
chus, physician to Amyntas, King of Macedon. His parents 
dying during his childhood, he was brought up by Proxenus 
of Atarna in Mysia, and at the age of seventeen became the 
disciple of Plato, who used to call him 'the mind of his school.'" 

(Gorton's Biographical Dictionary, v. I.) 

1060. "Proclus (A. D. 409) endeavored to change the entire 
framework of human reason; but his logical views are so inti- 
mately blended with his theology, that we can scarcely sepa- 
rate them for a special notice. He cultivated the Greek logic, 
but founded upon it the Eastern ideas of illumination or intui- 
tion ; and this led to almost impenetrable darkness and mysti- 
cism. The human mind, according to Proclus, may be viewed 
under two great categories, — identity and diversity." 

(Blakey's Hist. Sketch of Logic, p. 102.) 

1061. The analytical process, which formed such a conspic- 
uous ingredient in the Socratic logic, was nothing more or 
less than an exhibition of that inward movement, which every 
man of sane mind, no matter what portion of acquired know- 
ledge he may possess, carries on almost every moment of his 
life. Our minds are perpetually dividing the aggregate rep- 
resentations of things presented to its contemplation, whether 
of a physical or mental stamp, and resolving them, as it were, 
into their original or primary elements; and after this is 
effected, we sum them all up again, contemplate the represen- 
tations as entire and perfect wholes or compound conceptions, 
and fix them as such in the mind. This mental process is so 
subtle and rapid, that we seldom can arrest the trains of 
thought which constitute it, a sufficient length of time to bring 
the faculty of attention to bear upon and observe them. 

(lb. p. 19.) 

1062. "Sextus Empericus (supposed to have flourished in 
the reign of the Emperor Commodus, A. D. 161). He taught: 
All categories such as genus and species are useless, one-sided, 
imperfect and often completely false. If we consider them 
as purely mental conceptions or controversies of the mind, how 
can we determine their relation to external things? For any- 
thing we know to the contrary, the mental instrument may 
have no real or true relation, whatever, to the thing on which if 
operates." 

(lb. p. 106.) 



§§ 1063-1067.J HISTORICAL NOTES. 609 

1063. "The schoolmen is a name given to the leaders of 
thought in the scholastic period. The most eminent were: 
Johannes Scotus Erigena, (died circ. 886.) Anselm, Archbishop 
of Canterbury, (1033-1109), William of Champeaux, (died 1121), 
Peter Lombard, (died 1164), Alexander of Halles, (died 1245), 
St. Bonaventure, (died 1274), Albertus Magnus (1193-1280), 
St. Thomas Aquinas (circ. 1225-74), Duns Scotus (died 1308), 
Buridan (died after 1350), and Johannes Gerson, who endeav- 
ored to combine mysticism with scholasticism (1363-1429)." 

(Encyclopedic Dictionary.) 

1064. "Another striking feature of the schoolmen is their 
incessant and pertinaceous disputes on the nature of particu- 
lar and universal ideas. This is one of the most conspicuous 
incidents in their history, and has served alike to hand down 
their fame to posterity and to make them, in the eyes of many, 
objects of commiseration and contempt. 

For the sake of those who may not know the general merits 
of the question, we shall make a few explanatory observations 
upon it." 

(Blakey's Hist. Sketch of Logic, p. 128.) 

1065. "The point of dispute is simply thisj — the Nominal- 
ists affirm that there are two classes of truths; one class rela- 
ting to individual or single objects and their particular quali- 
ties or properties; the other class to general collections or 
assortments of things, which we designate by a general term 
or terms. A man is a particular idea; a multitude of men, a 
general idea. The Nominalists affirm that the difference be- 
tween those two kinds of ideas is only a verbal one, i. e., when 
men talk or reason about these general ideas or attributes of 
things, the general term is the only thing with which the mind 
is conversant." 

(lb. p. 128.) 

1066. "Now, the Kealists denied this doctrine in toto. They 
maintained that though these general terms are used in our 
descriptions of the similar properties or qualities of things, 
yet, there is a general idea always present in the mind, when 
it thus characterizes the common attributes which belong to 
a particular genus or class. This general term is not a mere 
verbal instrument or symbol, but stands for a real permanent 
intellectual conception, which is always present to the mind 
and to which the name of general idea is uniformlv given." 

(lb. p. 128.) 

1067. "Some reasoners attempted to steer a middle course; 
they were called Conceptualists. They agreed with the Nom- 
inalists in denouncing general ideas or conceptions, as the 
Realists considered them to be; but they still thought the mind 
had the power of creating those general ideas, which they pre- 

39 



610 APPENDIX. [ §§ 1068-1070. 

ferred to call conceptions. They said there were no essences 
or universal ideas to agree with general terms, and that the 
mind could reason about classes of individuals without the 
mediation of language." 

(lb. p. 129.) 

1068. "It may be observed in passing, that the schoolmen 
must not be considered as the originators of this controversy 
about particular and universal ideas. We can trace it in the 
oldest records we have of logical philosophy. Plato, Aris- 
totle, the Stoics and many other philosophers and sects, entered 
deeply into the entire question. They were all, however, un- 
able to solve it, and it descended down to the schoolmen of 
the Middle Ages, with all its puzzling freshness and inherent 
mysterv." 

(lb. p. 129.) 

1069. "Logic or dialectics, enjoyed great esteem in later 
times, particularly in the Middle Ages, so that it was consid- 
ered almost as the spring of all science, and was taught as a 
liberal art from the eighth century. The triumph of logic 
was the scholastic philosophy, which was but a new form of 
the ancient sophistry, and theology, particularly, became filled 
with verbal subtleties. 

Raymond Lully strove to give logic another form. The 
scholastics were attacked by Campanella, Gassendi, Peter 
Ramus (Pierre de la Ramee), Bacon, and others, with well 
founded objections." 

(Encyclopedia Americana, vol. 8, p. 49.) 

1070. Roscellinus, A. D. 1089. "This scholastic was canon 
of Compeigne. He is commonly considered as the first writer 
who distinctly broached the Nominalist theory. He main- 
tained that all general terms or names used in formal proposi- 
tions, are but simple mental abstractions which the mind forms 
by comparing a certain number of individuals with each other. 
In fact, he went the full length of maintaining that universals 
were nothing but names. 

This position appeared novel and startling to his age and 
hence it was that he drew upon himself ecclesiastical censure 
and rebuke. 

Roscellinus was obliged to retract his opinions at the council 
of Soissons, held in the year 1092. He was afterwards ban 
ished both from England and France. The theological bear- 
ings of the logical questions were the real cause of his defeat 
and punishment. He taught, "Tres persona s esse tres reali- 
tates differentes," a proposition, says his antagonist St. 
Anaelm, that ought to warn every one how cautiously they 
should handle questions of Holy Writ." 

(Blakey's Hist. Sketch of Logic, p. 134.) 



U 1071, 1072.J HISTORICAL NOTES. 611 

1071. Pierre Abelard, A. D. 1142. "The name of Abelard 
is intimately connected with the early history of scholastic 
logic. He was a zealous Nominalist and jealously contended 
for the validity of his theory through every phase of his event- 
ful life." 

John of Salisbury says: "that Abelard and his disciples 
looked upon the proposition that we can affirm one thing from 
another thing, as a great absurdity, though this absurdity was 
backed by the authority of Aristotle.' 

(lb. p. 140.) 

1072. Raymond Lully, A. D. 1309. 

"This was a zealous but eccentric logician. His life in con- 
nection with logical and philosophical studies is full of roman- 
tic interest. His Ars Magna is the exposition of a plan to 
enable the mind to work out all kinds of propositions through 
£he means of a mechanical table of ideas, disposed in such a 
manner that their different correlations would furnish satis- 
factory answers to every imaginable sort of questions. A 
great deal of ingenuity is displayed in this logical scheme; and 
some degree of interest was at first excited in different schools 
of learning, as to its practical and successful application. 
But its barrenness and formality soon became apparent, and 
many of the scholastic doctors pronounced it as useless, and 
as little better than a severe satire upon the entire system of 
dialectic mechanism. During the life of Lully, and for nearly 
two centuries after his death, his opinions on logical science 
were pretty generally adopted in seminaries of learning, both 
in Majorca and in a part of Spain. Even in the colleges of 
Parma, Montpelier, Paris and Rome, he was cordially 
esteemed as a logician whose general views were both enlight- 
ened and highly favorable to sound religion and morality. 
His theory of reasoning was nearly, in all cases, however, 
adopted with some reservations; and he was admired more for 
his ingenuity than for soundness and comprehensiveness of 
judgment. The doctors of the Sorbonne protested against the 
system of Lully, although it was taught with great eclat at 
Toulouse by Raymond de Sebonde. Politian praises his 
method; and Leibnitz himself, thought his logical works a 
monument of genius and industry. He has been alike the 
object of ardent admiration and severe censure. Whilst it 
has been declared that the simple touch of his handkerchief 
frequently cured hundreds of the sick, yet the Church, at one 
time, pronounced himself and all his disciples as heretics, and 
Gregory IX placed his writings, by a formal bull, in the Index 
Expurgatorius. There seems to have been so much vitality in 



612 APPENDIX. [ §§ 1073-1075. 

his system as to maintain its remembrance for a considerable 
time after the death of its founder." 

(lb. p. 154.) 

1073. William Occam, A. D., 1320. "Occam was a native 
of the county of Kent, studied at Merton College, Oxford, 
under the celebrated Duns Scotus, and was called the Invin- 
cible Doctor. The Kealistic doctrines met with a bold and 
formidable opponent in Occam. He adopted a certain form 
of the Nominalists' theory. He maintained that general ideas 
could not have an existence independent of external things, 
and of the Deity * * * "Every substance," says he, "is 
numerically one and singular; it is itself, and no other. It is 
not the same with a universal. If the universal were a thing 
existing in a number of individual or particular things, it 
would then possess a distinct and independent existence; for 
everything which is superior to another thing, must, according 
to the established laws of God, be independent of that thing, — 
a consequence which leads to a gross absurdity in reference 
to universal notions." * * * The commentators and crit- 
ics of Occam have been by no means agreed as to the precise 
nature of his own opinions. He is charged with arguing in 
the most decided manner against the Realists, stating the 
case of the Nominalists, and then leaving the question without 
offering his own opinions upon it. What these are, really 
seem to be that he could not go the whole length with the 
Nominalists' theory and that he was substantially what is 
denominated a Conceptualist." 

(lb. p. 156.) 

1074. Peter Ramus, A. D. 1515. "The leading notion 
which seemed to have occupied the mind of Ramus relative to 
logic was, that all its formal rules should be pure transcripts 
of the laws of thought as these are displayed in the act of 
reasoning. Nothing should be admitted into any system that 
will not bear this test. He denned logic to be the art of dis- 
coursing correctly or justly; and the examples which he gives 
are chieflv taken from the ancient orators and poets." 

(lb. p. 172.) 

1075. Thomas Hobbes, b. 1588, d. 1679. "The universality 
of one name to many things, has been the cause that men think 
the things are themselves universal, and so seriously contend 
that besides Peter and John and all the rest of the men that 
are, have been, or shall be, in the world, there is yet something 
else that we call man, viz.: man, in general, deceiving them- 
selves by taking the universal or general appellation for the 
thing it signifieth." "Logic is," he says, "the art of computa- 
tion." "Logicians add together two names to make an affir- 
mation, and two affirmations to make a syllogism, and many 



U 1076, 1077.] HISTORICAL NOTES. . 613 

syllogisms to make a demonstration; and from the sum or 
conclusion of a syllogism they subtract one proposition to find 
another." "Keason is nothing but reckoning, i. e., adding and 
subtracting of the consequences of general names agreed upon, 
for the marking and signifying of our thoughts." 

In the author's Logica we find the same doctrine main- 
tained. "An universal," says he, "is not a name of many 
taken collectively, but of each thing taken separately. Man 
is not the name of the human family in general, but of each 
single member of it, as Peter, John, and the rest separately. 
Therefore, this universal name is not the name of anything 
existing in nature, or of any idea or fantasm formed in the 
mind, but remains so by some word or name. It thus happens 
that when an animal or a stone or a ghost, or anything else, 
is called universal, we are not to understand by this term that 
any man or stone, or anything else, was or is or can be an 
universal; but only that these terms, animal, stone, and the 
like, are universal names, i. e., names common to many things; 
and the ideas or conceptions corresponding to them in the 
intellect, are the images or fantasms of single animals or other 
things, and, consequently we do not need, in order to com- 
prehend what is meant by an universal, any other faculty than 
that of imagination, by which we remember that such words 
have excited the ideas in our mind, sometimes of one particu- 
lar thing, sometimes of another. 

If speech be peculiar to man, as for aught I know it is, 
then is understanding peculiar to him also; understanding 
being nothing else but conception caused by speech. True 
and false are attributes of speech, not of things; where speech 
is not there is neither truth nor falsehood, though there may 
be error. Hence, as truth consists in the right ordering of 
names in our affirmations a man that seeks precise truth hath 
need to remember what every word he uses stands for and 
place it accordingly." 

(lb. p. 227.) 

1076. Jacques Benigne Bossuet, born in 1627, died 1704. 
"He defines truth to be that which exists, and falsehood that 
which has no existence. Truth being eternal, it must of neces- 
sity rest upon Deity. All necessary truths and principles 
existed prior to the human understanding; and consequently 
we can only be said to find truths, not to create them." 

(lb. p. 268.) 

1077. John Locke, born 1632, died 1704. "God has not 
been so sparing to men to make them barely two-legged crea- 
tures, and left it to Aristotle to make them rational, i. e., those 
few of them that he can get to examine the grounds of syllo- 
gisms, as to see, that in about three score ways that three 



614 APPENDIX. [ §§ 1078-1080. 

propositions may be laid together, there are but fourteen 
wherein one may be sure that the conclusion is right. God 
has been more bountiful to mankind. He has given them a 
mind that can reason without being instructed in the methods 
of syllogism.' 7 

(lb. p. 276.) 

1078. Christian Wolff, born 1679, died 1754. "It was a 
favorite opinion of Wolff's that all our reasonings could be 
greatly facilitated by having recourse to a uniform system 
of signs. He conceived that hieroglyphical emblems or fig- 
ures might be so applied as to represent fully and forcibly all 
general notions and propositions." 

(lb. p. 288.) 

1079. Gottf. Ploucquet, born 1716, died 1790. "In his 
Methodus Calculandi in Logicis, and other works,, labored hard 
to introduce new elements into the science of logic. His great 
aim was to reduce all human knowledge to one or two simple 
principles or rules, and to establish upon these a logical 
method which would mechanically, as it were, convey know- 
ledge on every branch of science with infallible certainty and 
great expedition. Reasoning was to be reduced to its simple 
elements, and by means of algebraical signs rendered a matter 
of pure calculation. 

Logic was only, according to Ploucquet, the art of deducing 
by an immutable rule, the known from the unknown, and this 
is amply sufficient for the explanation of every department of 
human inquiry. He reduces all judgments on facts or experi- 
ence to identical propositions, by the aid of the principle of 
sufficient reason." 

(lb. p. 293.) 

1080. Denys Diderot, born 1713, died 1784. "In many 
parts of his philosophical writings Diderot reduces reasoning 
to a mere species of sensation. He says, 'Every idea must 
necessarily, when brought to its state of ultimate decompo- 
sition, resolve itself into a sensible representation or picture; 
and, since everything in our understanding has been intro- 
duced there by the channel of sensation, whatever proceeds 
out of the understanding is either chimerical or must be able 
in returning by the same road, to reattach itself to its sensible 
archetype. Hence, an important rule in philosophy, that 
every expression which cannot find an external and a sensible 
object to which it can thus establish its affinity, is destitute of 
signification.' Helvetius affirms, likewise, that all truths may 
be reduced to simple facts, or identical propositions, A = B. 
The reasoning process is nothing more, he says, than the devel- 
opment of this simple law of our intellectual existence." 

(lb. p. 316.) 



§§ 1081-1084.] HISTORICAL NOTES. 615 

1081. Etienne Bonnot de Condillac, born 1715, died 1780. 
"Id Condillac's Logic he conceives that all reasoning may be 
ultimately resolved into the same form and certainty as mathe- 
matical evidence. The mode of accomplishing this would 
be to effect such improvements in language as to make it 
represent certain fixed and determined ideas. On the char- 
acter of the logic of Condillac, the author of his life and writ- 
ings in the last edition of the Encyclopedia Britannica, gives us 
the following opinion: he considers the justness of our reason- 
ings as depending on the degree of perfection of the language 
we possess. The superior certainty of mathematical as com- 
pared with other knowledge, is ascribed by him to the superior 
certainty of mathematical language." 

(lb. p. 318.) 

1082. George Campbell, born 1719, died 1796. "Dr. Camp- 
bell says the method of proving by syllogism appears, even on 
a superficial review, both unnatural and prolix. The rules 
laid down for distinguishing the conclusive from the incon- 
clusive forms of argument, the true syllogism from the var 
ious kinds of sophism, are at once cumbersome to the mem- 
ory and unnecessary in practice. No person, one may venture 
to pronounce, will ever be made a reasoner who stands in need 
of them. In a word, the whole bears the manifest indications 
of an artificial and ostentatious parade of learning, calculated 
for giving the appearance of great profundity to what, in 
fact, is very shallow. Such, I acknowledge, have been for a 
long time, my sentiments on the subject. On a mere inspec- 
tion I cannot say I have found reason to alter them, though 
I think I have seen a little further into the nature of this 
disputa»tive science, and, consequently, into the grounds of 
its futility." 

(lb. p. 348.) 

1083. Lord Karnes (Henry Homes) b. 1696, d. 1782.) 
"Lord Karnes was another Scottish writer who spoke lightly 

of the school logic. Speaking of reasoning in general, his lord- 
ship says, that all real knowledge of mankind may be divided 
into two parts, the first consisting of self-evident propositions; 
the second, of those which are deduced by just reasoning 
from self-evident propositions. The line which divides these 
two parts ought to be marked as distinctly as possible, and 
the principles that are self evident, reduced as far as can be 
done, to general axioms." 

(lb. p. 351.) 

1084. L. H. Wagner, 1806. 

"In his Logic, views the science of reasoning in a different 



616 APPENDIX. [ §§ 1085-1087. 

light from his contemporaries. His aim is to give a purely 
mathematical form to all logical rules, much after the same 
fashion as Lully and Bruno." 

(lb. p. 391.) 

1085. George Wilhelm Frederick Hegel, 1816. 

"In his Wissenschaft der Logik, denies that logic is merely 
expressive of the forms of thought; it constitutes its very 
essence and reality. Logic displays three different states or 
conditions. We simply consider and look at a thing. We 
then separate that thing from others, for nothing can exist in 
absolute unity; it must have two aspects and then out of these 
arises a certain relation which alone constitutes truth, reality 
being the absolute." 

(lb. p. 391.) 

1086. "Ventura (1828) enters profoundly into the science of 
method. All logical methods proceed on the principle of 
analysis. The mind looks at an entire system or a large assem- 
blage of general principles, and then seems to set about the 
work of analysis or separation with a view of realizing one 
general idea, which is known only to itself and which is often 
obtained by a mental process, which entirely eludes the most 
searching efforts of consciousness. Everything must be taken 
to pieces; every corner and crevice of the system must be 
examined before the several parts can be put together and 
adjusted agreeably to the scientific idea which we have in our 
own minds, and which we set out in our inquiries to establish 
and realize. These are the leading steps of the mental process 
in every philosophical method." 

(lb. p. 409.) 

1087. "Sir William Hamilton succeeded Dr. Ritchie in 1836 
in the Edinburg University. About four years after this, it 
is said that the Professor introduced what is termed his new 
analytic method of teaching formal logic. This method pro- 
ceeds on a thoroughgoing quantification of the predicate. By 
the adoption of this principle we are told that past evils are 
corrected, past omissions supplied and logic receives its high- 
est development in the perfection and simplicity of its form." 

"The entire doctrine of the conversion of syllogisms is, (on 
the principle of this new analytic method of Sir William) pro- 
nounced to be useless and false. This inconsistent and cum- 
berous doctrine resulted, as we have said, from a false analysis 
by logicians of the elements with which they had to deal. The 
whole doctrine is founded upon the relation of quantity 
between the subject and predicate in a proposition; but if a 
principle element of that relation be left out, the doctrine will 
of course be defective. Logicians stand chargeable with this 
neglect. They commenced to recompose their system before, 



§§ 1088, 1089.] HISTORICAL NOTES. 617 

by thorough decomposition, they had obtained all the elements 
requisite for that system." 

(lb. p. 445.) 

1088. When Dr. Thomas Brown's lectures on the Philoso- 
phy of the Human Mind (1822) made their appearance, a new 
direction was given to mental science. "What is it which 
the syllogistic art would confer on us in addition? To shorten 
the process of arriving at truth, it forces us to use, in every 
case, three propositions instead of the two which nature directs 
us to use. Instead of allowing us to say: Man is fallible, he 
may therefore err, even when he thinks himself most secure 
from error, — which is the spontaneous order of analysis in 
reasoning, — the syllogistic art compels us to take a longer 
journey to the same conclusion by the use of what it calls a 
major proposition, — a proposition which never rises spontan- 
eously, for the best of all reasons, that it cannot rise without 
knowledge of the very truth which is by supposition unknown. 
To proceed in the regular form of a syllogism, we must say, 
All things that are fallible may err, even when they think 
themselves most secure from error. But man is a fallible 
being, he may therefore err, even when he thinks himself most 
secure from error. 

In our spontaneous reasonings, in which we arrive at pre- 
cisely the same conclusions, and with a feeling of evidence 
precisely the same, there are, as I have said, no major propo- 
sitions, but simply what in this futile art are termed techni- 
cally the minor and the conclusion. The invention and formal 
statement of a major proposition then, in every case, serve 
only to retard the progress of discovery, not to quicken it or 
render it in the slightest degree more sure. 

Again, he observes, the syllogism, therefore, which proceeds 
from the axiom to the demonstration of particulars, reverses 
completely the order of reasoning and begins with the conclu- 
sion, in order to teach us how we may arrive at it." 

(lb. p. 452.) 

1089. John Stuart Mill, b. 1806, d. 1873. 

"It must be granted," says Mr. Mill, "that in every syllogism 
considered as an argument to prove the conclusion, there is a 
petitio princippii. When we say, — All men are mortal; Soc- 
rates is a man, therefore, Socrates is mortal; it is unan- 
swerably urged by the adversaries of the syllogistic theory 
that the proposition, Socrates is mortal, is pre-supposed in 
the more general assumption, All men are mortal; that we 
cannot be assured of the mortality of all men, unless we were 
previously certain of the mortality of every individual man; 
that if it be still doubted whether Socrates or any other indi- 
vidual you choose to name be mortal or not, the same degree 



618 APPENDIX. [ §§ 1090, 1091. 

of uncertainty must hang over the assertion, All men are 
mortal; that the general principle, instead of being given as 
evidence of the particular case, cannot itself be taken for true, 
without exception, until every shadow of doubt which could 
affect any case comprised within it is dispelled by evidence 
aliunde; and then what remains for the syllogism to prove? 

That, in short, no reasonings from generals to particulars 
can, as such, prove anything, since from a general principle 
we cannot infer any particulars but those which the principle 
itself assumes as foreknown. This doctrine is irrefragable." 

(lb. p. 467.) 

1090. George Boole, d. 1864. 

"The time must come when the inevitable result of the ad- 
mirable investigations of the late Dr. Boole must be recog- 
nized at their true value, and the plain and palpable form in 
which the machine presents those results will, I hope, hasten 
the time. Undoubtedly Boole's life marks an era in the sci- 
ence of human reason. It may seem strange that it had 
remained for him first to set forth in its full extent, the prob- 
lem of logic, but I am not aware that anyone before him had 
treated logic as a symbolic method for evolving from any pre- 
mises the description of any class whatsoever, as defined by 
these premises. In spite of several serious errors into which 
he fell, it will probably be allowed that Boole discovered the 
true and general form of logic, and put the science substau- 
tially into the form which it must hold for evermore. He thus 
effected a reform with which there is hardly anything com- 
parable in the history of logic between his time and the remote 
age of Aristotle. Nevertheless, Boole's quasi-mathematical 
system could hardly be regarded as a final and unexception- 
able solution of the problem. Not only did it require the man- 
ipulation of mathematical symbols in a very intricate and per- 
plexing manner, but the results when obtained were devoid of 
demonstrative force because they turned upon the employ- 
ment of unintelligible symbols, acquiring meaning only by 
analogy." 

(Jevon's Principles of Science, p. 113.) 

1091. Archbishop Richard Whately, b. 1789, d. 1863. 

"In his work on Logic, pp. 82-86, makes some excellent 
remarks on probabilities. He says, 'But though when one pre- 
mise is certain, and the other only probable, it is evident that 
the conclusion will be exactly as probable as the doubtful pre- 
mise, and there is some liability to mistake in cases where each 
premise is merely probable. For, though most every one would 
perceive that in this case, the probability of the conclusion 
must be less than that of either premise, the precise degree 
in which its probability is diminished is not always so readily 



I 1091.] HISTORICAL NOTES. 619 

apprehended. And yet this is a matter of exact and easy 
arithmetical calculation. I mean, that given the probability 
of each premise, we can readily calculate, and with perfect 
exactness, the probability of the conclusion. As for the prob- 
ability of the premises themselves, that are put before us, 
that of course must depend on our knowledge of the subject- 
matter to which they relate. But supposing it agreed what 
the amount of probability is in each premise, then, we have 
only to state that probability in the form of a fraction, and to 
multiply the two fractions together, the product of which will 
give the degree of probability of the conclusion. Let the prob- 
ability, for instance, of each premise be supposed the same, 
and let it in each be 2-3, (that is, let each premise be supposed 
to have two to one in its favor, that is, to be twice as likely to 
be true as to be false), then the probability of the conclusion 
will be two-thirds of two-thirds, 4-9; — rather less than one-half. 
For, since twice two is four, and thrice three nine, the frac- 
tion expressing the probability of the conclusion will be 
four-ninths. When you have two (or more) distinct arguments 
each separately establishing as probable the same conclusion, 
the mode of proceeding to compute the total probability is 
the reverse of that mentioned just above, for there, in the case 
of two probable premises, we consider what is the probability 
of their being both true, which is requisite in order that the 
conclusion may be established by them. 

But in the case of a conclusion twice (or oftener) proved 
probable by separate arguments, if these indications of truth 
do not all of them fail, the conclusion is established. You con- 
sider, therefore, what is the probability of both of these indi- 
cations of truth being combined in favor of any conclusion 
that is not true. Hence, the mode of computation is to state 
(as a fraction) the chances against the conclusion as proved by 
each argument, and to multiply these fractions together to 
ascertain the chances against the conclusion as resting on both 
the arguments combined, and this fraction being subtracted 
from unity, the remainder will be the probability for the con- 
clusion. For instance, let the probability of a conclusion as 
established by a certain argument, be 4-9: (suppose that this 
man is the perpetrator of a certain murder, from stains of 
blood being found on his clothes), and again, of the same con- 
clusion as established by another argument, 2-5 : (suppose from 
+he testimony of some witness of somewhat doubtful charac- 
ter), then, the chances against the conclusion in each case 
respectively, will be 5-9 and 3-5, which multiplied together 
give 15-45ths or 1-3 against the conclusion. The probability, 
therefore, for the conclusion as depending on these two argu- 



620 APPENDIX. [ §§ 1092, 1093. 

ments jointly, (i. e., that he is guilty of the murder) will be 2-3, 
or two to one. 

As for the degree of probability of each premise, that, as we 
have said, must depend on the subject matter before us, and it 
would be manifestly impossible to lay down any fixed rules for 
judging this. But it would be absurd to complain of the want 
of rules determining a point for which it is plain no precise 
rules can be given, or to disparage for that reason, such rules 
as can be given for the determining of another point. 

Mathematical science will enable us, given one side of a tri- 
angle and the adjacent angles, to ascertain the other sides, 
and this is acknowledged to be something worth learning, 
although mathematics will not enable us to answer the ques- 
tion which is sometimes proposed in jest, of "How long is a 
rope?" 

1092. In Brewster's Encyclopedia there is an article on 
Logic, from which I make the following quotations : 

"The name of Aristotle is inseparably connected with the 
history of Dialectics. He claims the sole honor of the inven- 
tion of syllogism; and his claims have been generally recog- 
nized; for, though the syllogistic mode of reasoning had been 
long known and much practiced before his time, yet, he had the 
merit of reducing to a system, those principles which, till then, 
had lain as a chaos of undigested materials. The utility of 
his labors, indeed, has not only been disputed in modern times, 
but they have been denounced as the chief cause of retardation 
in the progressive march of the human intellect; yet no one 
ever doubted the ingenuity which planned, and the industry 
which perfected a system which exerted a predominant influ- 
ence over the human mind for many ages." 
(B. E. XIII, p. 142.) 

1093. "But the system, by whomsoever raised, is a stupen- 
dous monument to misapplied ingenuity. It owed its attrac- 
tion to its high pretensions, for there is something very impos- 
ing in the idea of possessing a rule by which we may measure 
the pretensions of every doctrine and subject to the dominion 
of our own mind, every branch of human knowledge. We need 
not be surprised then, that Aristotle, the real or the supposed 
inventor of this art, should be regarded with veneration and 
considered as a benefactor to the human race. 

It was long before mankind could be induced to doubt the 
infallibility of their guide, till at last his influence was shaken 
by experience of the inefficacy of his method for the discovery 
of truth, and by the splendid labors of a few, who dared to 
abandon the intellectual weapons of dialecticians and trust 
their fame to a humble investigation of the laws of nature." 

(lb. p. 142.) 



§§ 1094-1096.] HISTORICAL NOTES. 621 

1094. "We once witnessed an instance of this confusion of 
ideas in a court of justice. The judge thus questioned a wit- 
ness — Did you hear A say that B had given him a blow which 
would be his death? The witness answered, No. Did you hear 
A say that B had not given him a blow? To this the witness 
also answered, No. The judge not perceiving that his evidence 
implied no contradiction, threatened to commit him to prison 
for perjury and equivocation." 

(lb. p. 145.) 

1095. "Such has been the revolution of philosophical opin- 
ion, that syllogism, which for so many centuries had been con- 
sidered as the grand bulwark of reasoning, is now almost uni- 
versally exploded; and any man who should digest his argu- 
ments into a syllogistic form, or who should say a word in 
favor of the practice, would excite ridicule rather than pro- 
duce conviction. 

The authority of Bacon, of Locke or Keid, and of Stewart, 
(not to mention hundreds of their followers, who, under their 
protecting shield, have furiously attacted a doctrine which 
they never attempted to comprehend), would of itself, have 
been sufficient to shake the credit of a system to which the 
philosophical world had so long bowed with submission." 

(lb. p. 147.) 

1096. Victor Cousin, b. 1792, d. 1867. 

"Cousin wrote a very interesting work called "History of 
Mental Philosophy." From it I take the following selections: 

"'The inquiry that we are about to make in order to be 
methodically directed, should be divided into three points. 
First, it is necessary to state and enumerate in their integrity, 
the elements or essential ideas of reason. We must have them 
all and be sure, at the same time, that we suppose none and 
that we omit none; for, if we imagine a single one, a hypothet- 
ical element would lead us to hypothetical relations and thence 
to a hypothetical system. The first law of a wise method is 
then, a complete enumeration. 

The second is an examination so profound, of all these ele- 
ments, that it may result in their reduction and that we may 
finish by having in hand the determinate number of elements, 
simple and indecomposible, which form the boundary of analy- 
sis. 

The third law of method is the examination of the different 
relations of these elements among themselves. I say the dif- 
ferent relations, for these elements may sustain a great num- 
ber of different relations. None must be supposed, nor must 
be neglected. It is when we shall have all these elements, 
when we shall have reduced them, when we shall have seized 



622 APPENDIX. [§§ 1097, 1098. 

upon all their relations, that we shall be in possession of the 
foundations of reason and of its history." 

(V. C. Ment. Philos. v. I, 66-67.) 

1097. "I speak to you of unity, you cannot avoid thinking of 
variety. When I speak to you of the infinite you cannot avoid 
conceiving the finite. We must not say, as is said by two 
great rival schools, that the human mind begins by unity and 
the infinite ; or by the finite, the contingent and the multiple ; 
for, if it begins by unity, I defy it ever to arrive at multiplicity; 
or, if it starts at multiplicity alone, I defy it equally to arrive 
ever at unity; if it starts from phenomena alone, it would not 
arrive at substance; if it starts from the idea of imperfection 
it would not arrive at perfection; and reciprocally. 

The two fundamental ideas to which reason is reduced, are 
then, two contemporaneous ideas. The one supposes the other 
in the order of the acquisition of our knowledge. As then, we 
do not begin only by the senses and experience, and as we do 
not any more begin by abstract thought and by intelligence 
alone, so the human mind begins neither by unity nor by 
multiplicity; it begins and cannot avoid beginning by both; 
the one is the opposite of the other, a contrary implying 
its contrary; the one exists only on condition that the other 
exists at the same time. Such is the order of the acquisition of 
our knowledge. The order of nature is different." 

(lb. p. 70.) 

1098. "Call to mind the conclusions of the last lecture. 
Eeason, in whatever way it may occupy itself, can conceive 
nothing except under the condition of two ideas which preside 
over the exercise of its activity: idea of the unit and of the 
multiple, of the finite and of the infinite, of being and of appear- 
ing, of substance and of phenomenon, of the absolute cause 
and of secondary causes, of the absolute and of the relative, 
of the necessary and of the contingent, of immensity and of 
space, of eternity and of time, etc. 

Analysis in bringing together all these propositions, in bring- 
ing together, for example, all their first terms, identifies them; 
it identifies equally, all the second terms; so that of all of these 
propositions, compared and combined, it forms a single propo- 
sition, a single formula, which is the formula itself, of thought 
and which you can express, according to the case, by unit and 
by the multiple, the absolute being and the relative being, unity 
and variety, etc. 

Finally, the two terms of this formula, so comprehensive, 
do not constitute a dualism in which the first term is on one 
side, the second on the other, without any other relation than 
that of being perceived at the same time by reason." 

(lb. p. 73.) 



§§ 1099-1104.] HISTORICAL NOTES. 623 

1099. "Reason conceives a mathematical truth ; can it change 
this conception as my will changed just now my resolution? 
Can it conceive that two and two do not make four? Try and 
you will not succeed; and not only in mathematics but in all 
the other spheres of reason, the same phenomenon takes place. 
In morals, try to conceive that the just is not obligatory; in 
art try to conceive that such or such a form is not beau- 
tiful ; you will try it in vain, reason will always impose upon 
you the same conception. Reason does not modify itself ac- 
cording to our taste; you do not think as you wish." 

(lb. pp. 75-76.) 

1100. "The necessary condition of intelligence is conscious- 
ness, that is difference." 

(lb. p. 77.) 

1101. "At the bottom of everv negation lies an affirmation." 

(lb. p/ 81.) 

1102. From Herbert Spencer's great work. "Principles of 
Psychology," vol. 2, I make the following quotations: 

"The syllogism then, if taken to represent the form of the 
inferential act, has the fundamental fault that it fails to cover 
the whole of the ground it professes to cover. It falls short at 
both ends. There are simple deliverences of reason and com- 
plex deliverences of reason, both of thern^ having the highest 
decree of certainty, which are entirely extra -svllogistic. cannot 
however, violentlv dislocated, be brought within the svllogistic 
form. Conseouently, if it be admitted that a true expression 
of the ratiocinative act must be one applicable to all ratiocin- 
ative acts, it must be concluded that the ratiocinative act is 
not trulv expressed bv the svlloe-ism." 

(lb. pp. 96-97.) 

1103. "Abilitv to perceive eonality implies a correlative 
ability to perceive ineoualitv: neither can exist without the 
other. But. though inseparable in origin, the cognitions of 
eonalitv and ineoualitv, whether between things or relations, 
differ in this, that while the one is definite, the other is indefi- 
nite. There is but one eonalitv, but there are numberless 
decrees of inequality. To assert an ineoualitv involves the 
affirmation of no fact, but merely the denial of a fact; and, 
therefore, as positing nothing" specific, the cognition of ine- 
oualitv can never be a premise to anv conclusion." 

(lb. p. 26.) ' 

1104. "Obviouslv, then, the process of thought formulated 
by the syllogism, is in various ways irreconcilable with the 
process of reasoning as normally conducted, irreconcilable 
as presenting the class, while vet there is nothing to account 
for its presentation; irreconcilable as predicating of that class 
a special attribute while yet there is nothing to account for 



624 APPENDIX. [ §§ 1105-1108. 

its being thought of in connection with that atttribute; irre- 
concilable as embodying in the minor premise, an assertory 
judgment (this is a man), while the previous reference to the 
class men, implies that that judgment had been tacitly formed 
beforehand; irreconcilable as separating the minor premise 
and the conclusion, which ever present themselves to the mind 
in relation." 

.(lb. p. 99.) 

1105. "The proposition, 'I have a pain,' may be called in 
contrast with most propositions a simple one; though even it 
involves the unexpressed propositions that I have a body, and 
this body has a part in which this pain is localized, and that I 
have before had pains with which I class this as like in general 
nature. Strictly speaking, no such thing exists as an abso- 
lutely simple proposition, implying nothing beyond one sub- 
ject and one predicate known in relation. Nevertheless, though 
the simplest proposition connotes sundry other propositions, 
there is a broad line to be drawn between it and the great mass 
of propositions, which severally make multitudes of predica- 
tions beyond that which they appear to make." 

(lb. pp. 395-6.) 

1106. "Clearly, then, that we can compare conclusions with 
scientific rigour, we must not only resolve arguments into their 
constituent propositions, but must resolve each complex propo- 
sition into the simple propositions composing it. And only 
when each of these simple propositions has been separately 
tested, can the complex proposition, made up of them, be 
regarded as having approximately, a validity equal with that 
of a simple proposition which has been tested." 

(lb. p. 399.) 

1107. 'Still, there rises the question, how are we to choose 
between opposing conclusions, each of which claims to be 
legitimately drawn from premises alleged to be beyond doubt? 
Arguments of all kinds, including those of metaphysicians, 
which we have here to value, proceed upon the tacit assump- 
tion that each datum and each successive step has that indu- 
bitable warrant, the nature of which we have been examining." 

(lb. pp. 428-29.) 

1108. "Two reasons may be distinguished for insisting on 
this testing process. One is, that in proportion as proposi- 
tions are compound, direct comparisons of them must be haz- 
ardous; because their component propositions, each of which is 
an inlet to possible error, cannot be severally tested and veri- 
fied. The other is, that only when compound propositions 
are resolved into their constituents, can it be seen what are the 



§§ 1109-1113.] HERBERT SPENCER. 625 

relative numbers of assumptions in the two, and what are the 
relative possibilities of error hence resulting.' 7 

(lb. p. 429.) 

1109. "The general notions of agreement and disagreement, 
apply equally to two lines compared in their lengths and to two 
accounts of an event ; and hence, in the absence of experiences 
that yield this general notion, accuracy of thought and pre- 
cision of statement are not possible." 

(lb. p. 530.) 

1110. "To say — This is an animal,' or 'This is a circle,' or 
'This is the color red,' necessarily implies that animals, circles 
and colors have been previously presented to consciousness. 
And the assertion that this is an animal, a circle, or a color, 
is a grouping of the new object perceived, with the similar 
objects remembered. In like manner, the inferences — 'That 
berry is poisonous,' 'This solution will crystalize,' are impos- 
sible, even as conceptions, unless a knowledge of the relations 
between poison and death, between solution and crystaliza- 
tion, have been previously put into the mind, either imme- 
diately bv experience or mediatelv bv description." 

(lb. pp. 114-115.) 

1111. "Thus, the belief in an unchanging order, the belief 
in law. now spreading among the more cultivated throughout 
the civilized world, is a belief of which the primitive man is 
absolutely incapable. Not simply does he lack the experiences 
that give materials for the conception, but he lacks the power 
of framing the conception; he is unable to think of a single law 
much less of law in general. The needful representativeness 
of thought is to be acquired only by the inherit ence of accumu- 
lated increments of faculty successively organized, and it ?s 
even now possessed in a high degree only by a very small 
minoritv." 

(lb. p. 529.) 

1112. From a fine article on Loeric in the Encyclopedia 
Britannica, vol. 14, written by Prof. Adamson, I make the fol- 
lowing extracts: 

"Even more radical is the divergence of modern logic from 
the Aristotelian ideal and method. The thinker who claimed 
for logic a special preeminence among sciences, because, "since 
Aristotle it has not had to retrace a single step, * * * and 
to the present dav has not been able to make one step in ad- 
vance," has himself, in his general modification of all philoso- 
phy, placed logic on so new a basis that the only point of con- 
nection rpfained bv it in his system, with the Aristotelian, 
may not be unfairly described as the community of subject." 

1113. "If. adopting a simpler method, one were to inspect a 
fair proportion of the more extensive recent works on Logic. 

40 



626 APPENDIX. [§§1114,1115. 

the conclusion drawn would probably be the same, — that, 
while the matters treated show a slight similarity, no more 
than would naturally result from the fact that thought is the 
subject analyzed, the diversity in mode of treatment is so 
great that it would be impossible to select by comparison and 
criticism a certain body of theorems and methods, and assign 
to them the title of logic. That such works as those of Tren- 
delenburg, Ueberweg, Ulrici, Lotze, Sigwart, Wundt, Berg- 
mann, Schuppe, DeMorgan, Boole, Jevons, and these are but a 
selection from the most recent, treat of notions, judgments 
and methods of reasoning, gives to them, indeed, a certain 
common character, but what other features do they possess 
in common? In tone, in method, in aim, in fundamental prin- 
ciples, in extent of field, they diverge so widely as to appear, 
not so many different expositions of the same science, but of so 
many different sciences. In short, looking to the chaotic 
state of logical text books at the present time, one would be 
inclined to say that there does not exist anywhere a recognized 
currently received body of speculations to which the title of 
logic can be unambiguously assigned, and that we must there- 
fore resign the hope of attaining by any empirical considera- 
tion of the received doctrine, a precise determination of the 
nature and limits of logical theory." 

(Enc. Br. p. 783.) 

1114. "Each perception is itself and is only itself; no judg- 
ment is possible save that of identity. In other words, if there 
be judgment at all, it can consist only in the assertion that the 
unanalyzed perception is identical with that into which it is 
analyzed, and as each perception and each analytic portion of 
a perception may be signified by an arbitrary sign (name or 
other hieroglyphic,) judgment is essentially an affair of naming, 
a declaration that different names are identical or belong to 
the same perception. 

Reasoning is simply the transition from identity to another — 
a more developed result of analysis. Scientific or real know- 
ledge, is an accurately framed system of signs, i. e., a collection 
of signs which expresses precisely the results of the analysis of 
complex perceptions. Logic, under this doctrine of know- 
ledge, is merely a statement of the various modes in which 
analysis is carried out ; of the ways in which names are applied, 
and of the forms in which names are combined. Such is tho 
theory of logic presented by Condillac." 

(lb. p. 794.) 

1115. "One development from the Psychology of Locke has 
thus appeared as an extreme formalism, which, if carried out 
consistently, must needs assume the aspect of a numerical or 
mechanicai system of computation. It is remarkable that 



§ 1116. J LEIBNITZ. 627 

a very similar result was reached by Leibnitz, a thinker who 
proceeded from a quite opposed psychological conception. 

The characteristics -of Scientia Generalis are at once deduc- 
ible from the two general principles, which in Leibnitz's view, 
dominate all our thinking, — the law of sufficient reason and the 
law of non-contradiction. It must contain a complete account 
of the modes in which, from data, conclusions are drawn, and 
in which, from given facts, data are inferred, and since the 
only logical relations are those of identity and non-contradic- 
tion, the forms of inference from or to data, must be the gene- 
ral modes of combinations of simple elementary facts which 
are possible under the law of non-contradiction. 

The statement of the data of any logical problem, and the 
description of the processes involved in combining them or 
arriving at them, are much assisted by, if not dependent on, 
the employment of a general characteristic or symbolic art. 

The fundamental divisions of Soientia Generalis, so far at 
least as its groundwork are concerned, (for Leibnitz sometimes 
includes under one head all possible applications of the theory,) 
are, (1) the synthetical or combinatorical art, the theory of the 
processes by which from given facts complex results may be 
obtained (of these processes which make up general mathesis, 
syllogistic and mathematical demonstrations are special vari- 
eties); (2) the analytic or regressive art, which starting from 
a complex fact, endeavors to attain knowledge of the data 
from whose combination it arose. 

Of the first art, the logical calculus in particular, a some- 
what clearer and fuller outline is given. The logical calculus 
implicated, 

(1) the statement of data in their simplest form, 

(2) the assignment of the general laws under which com- 

bination of these data is possible, 

(3) the complete exposition of the forms of combination, 

(4) the employment of a definite set of symbols, both of data 

and modes of combination, subject to symbolic laws 
arising from the laws under which combination is pos- 
sible." 

(lb. p. 794.) 
1116. "In the Fundamenta Calculi Ratiocinatoris and the 
Non-inelegans Speciman Demqnstrandi, something is effected 
toward filling up the first, second and fourth of these rubrics, 
but in no case is the treatment exhaustive. The simple data, 
called characters or formulae, are symbolized by letters, rela- 
tions of data by a somewhat complicated and varying system 
of algebraic signs; for the calculus, or set of operations exer- 
cised upon relations given, so as to produce a new formulae, 
no comprehensive system of symbols is adopted. 



628 APPENDIX. [§§1117-1119. 

Formulae, relations and operations take the place of notions, 
judgments and syllogism. The general laws of combination 
of data are stated without much precision. Leibnitz recog- 
nizes the law of substitution, notes also what have been called 
the laws of reduplication and commutativeness, but, in actual 
realization of his method, employs indifferently, the relation 
of containing and contained, of the relation of identical sub- 
stitution (gequippollence.) No attempt is made to develop 
a complete scheme of possible modes of combination. 

At the root of Leibnitz's universal calculus, as of Condillac's 
method of analysis, and generally of nominalist logic, there 
lies a peculiar acceptation of the abstract law of identity. 
That a thing is what it is, — that knowledge of a thing is a 
single, indivisible, mechanical fact, susceptible only of explic- 
ation or of expanded statement, — that is the principle domina- 
i ing logical theories which in other respects may differ widely. 
Insistence upon the aspect of knowledge or of the object known 
is the ground for assigning to thought a function purely analy- 
tic, which is the verv kevnote of nominalism." 

* (lb. p. 794.) 

1117. "Under all circumstances, difference is as important 
an element as identity, in the judgment, and to concentrate 
attention upon the identity, is to take a one-sided and imper- 
fect view. 

So soon, however, as the real nature of thought has been 
thrown out of account as not concerned in the processes of 
logic, so soon as the law of non-contradiction, in its manifold 
statement, has been formulated as the one principle of logical 
or formal thinking, there appears the possibility of evolving 
an exact svstem of the conditions of non-contradictoriness." 

(lb. p. 800.) 

1118. "The ultimate units of knowledge, whatsoever we call 
them, whether notions or ideas of classes of names, have at 
least one characteristic, — they are what they are, and there- 
fore, exclude from themselves whatever is contradictory of 
their nature. They are combined positions and negations, that 
which is posited or negated being left undetermined, — referred, 
in fact, to matter as opposed to form. With respect to any 
article of thought, therefore, the only logical requirement is 
that it shall possess the characteristic of not being self-con- 
tradictory, and the only logical question is, what exactly is 
posited and negated therebv." 

(lb. p. 800.) 
1110. "Hobbes' doctrine of thought as dealing with names 
and as essentially addition and subtraction of nameable feat- 
ures, Boole's doctrine of thought, as the determination of a 
«lass, Jevon's view of thought, as simple apprehension of 



§§ 1120-1122. J DeMORGAN. 629 

qualities, — any of these will serve as starting point, for in all 
of them the fruitful element is the same. 

The further step that the generalization of the system of 
thought must take a symbolic form, presents itself as an imme- 
diate and natural consequence." 

(lb. p. 800.) 

1120. "The first question which suggests itself in connection 
with Boole's symbolic logic, is the necessity or advisability of 
retaining the reference to classes, or the description of thought 
as classification. 

Do the symbolic laws really depend to any great extent on 
the logical peculiarities of class arrangement? Mr. Venn, who 
emphasizes this feature in Boole's scheme, has, however, done 
good service in leading up to a different explanation. The gen- 
eral reference to objects, which is also noted as implied in all 
Boole's formulas, has nothing to do with the possible difference 
of conceptualist or materialist doctrines of the proposition, 
and, in fact, as all distinctions of thing and quality, resem- 
blance and difference, higher and lower, subject and predicate 
vanish or are absorbed in the more general principle underly- 
ing the symbolic method, phrases such as classification, exten 
sion, intention and the like, should be banished as not perti- 
nent. Nay, the usual distinctions of quantity and even of qual- 
ity, either disappear or acquire a new significance, when they 
are brought under the scope of the new principle. What sym- 
bolic logic works upon by preference, is a system of dichotomy, 
of x and not x, y and not y and so forth. In other words, 
quantitive differences require to find expression through some 
combination of the positions and negations of the elements 
making up the objects dealt with, while the usual quantitive 
distinctions are merged in the position or negation of various 
combinations." 

(lb. p. 801.) 

1121. "There appears very clearly in Grassman's treatment 
the essence of the principle on which symbolic logic proceeds. 
Thought is viewed as simply the process of positing and negat- 
ting definite contents or units, and the operations of logic 
become methods for rendering explicit that which is in each 
case posited or negated. To apply symbolic methods, we 
require units as definite as those of quantitive science, and the 
only laws we can employ are those which spring from the 
nature of units as definite." 

(lb. p. 801.) 

1122. Augustus DeMorgan, b. 1806, d. 1871. From DeMor- 
gan's work on "Formal Logic," I make the following extracts: 

"Whether the premises be true or false, is not a question of 
logic, but of morals, philosophy, history, or any other knowl- 



630 APPENDIX. [ §§ 1123-1127. 

edge to which their subject-matter belongs: the question of 
logic is, Does the conclusion certainly follow if the premises be 
true?" 

(Formal Logic pp. 2, 4.) 

1123. "Every X is Y, affirms, Some X's are Y's, and denies 
No X is Y, Some X's are not Y's. 

No X is Y affirms Some X's are not Y's and denies Every X 
is Y, Some X's are Y's. 

Some X's are Y'ci does not contradict Every X is Y, Some 
X's are not Y's, but denies no X is Y. 

Some X's are not Y's does not contradict No X is Y, Some 
X's are Y's, but denies Every X is Y." 

(lb. pp. 4, 5.) 

1124. "Thus, the honest witness who said, 'I always thought 
him a respectable man — he kept his gig*' would probably not 
have admitted in direct terms, 'Every man who keeps a gig 
must be respectable.' " 

(lb. p. 20.) 

1125. "Thus, the idea of a horse is the horse in the mind: 
and we know no other horse. We admit that there is an 
external object, a horse, which may give a horse in the mind to 
twenty different persons : but no one of these twenty knows the 
object; each one only knows his idea." 

(lb. p. 29.) 

1126. "Connected with ideas are the names we give them ; 
the spoken or written sounds by which we think of them, and 
communicate with others about them. To have an idea and to 
make it the subject of thought as an idea, are two perfectly dis- 
tinct things : the idea of an idea is not the idea itself. I doubt 
whether we could have made thought itself the subject of 
thought without language. As it is, we give names to our 
ideas, meaning by a name not merely a single word, but any col- 
lection of words which conveys to one mind the idea in 
another. Thus, a-man-in-a-black-coat-riding-along-the-high- 
road-on-a-bay-horse, is as much the name of an idea as man, 
black, or horse. We can coin words at pleasure; and were it 
worth while, might invent a single word to stand for the pre- 
ceding phrase." 

(lb. 34.) 

1127. "Every name has a reference to every idea, either 
affirmative or negative. The term horse applies to every thing, 
either positively or negatively. This (no matter what I am 
speaking of) either is or is not a horse. If there be any doubt 
about it, either the idea is not precise, or the term horse is ill 
understood. A name ought to be like a boundary, which 
clearly and undeniably either shuts in or shuts out every idea 



§§ 1128-1130.] FORMAL LOGIC. 631 

that can be suggested. It is the imperfection of our minds, 
our language, and our knowledge of external things, that this 
clear and undeniable inclusion or exclusion is seldom attain- 
able, except as to ideas which are well within the boundary: at 
and near the boundary itself all is vague. 

There are decided greens and decided blues, but between the 
two colors there are shades of which it must be unsettled by 
universal agreement to which of the two colors they belong. 
To the eye, green passes into blue by imperceptible grada- 
tions: our senses will suggest no place on which all agree, at 
which one is to end and the other to begin. " 

(lb. p. 35.) 

1128. "When a name is clearly understood, by which we 
mean when of every object of thought we can distinctly say, 
this name does or does not contain that object — we have said 
that the name applies to everything, in one way or the other. 
The word man has an application both to Alexander and Bu- 
cephalus: the first was a man, the second was not. 

In the formation of language, a great niany names are, as to 
their original signification, of a purely negative character: 
thus, parallels are only lines which do not meet. Aliens are 
men who are not Britons (that is, in our country)." 

(lb. p. 37.) 

1129. "Let us take a pair of contrary names, as man and 
not-man. It is plain that between them they represent every- 
thing imaginable or real in the universe. But the contraries of 
common language usually embrace, not the whole universe, but 
some one general idea. Thus, of men, Briton and alien are con- 
traries : every man must be one of the two, no man can be both. 
Not-Briton and alien are identical terms, and so are not-alien 
and Briton. The same may be said of integer and fraction 
among numbers, peer and commoner among subjects of the 
realm, male and female among animals, and so on. In order to 
express this, let us say that the whole idea under consideration 
is the universe, (meaning merely the whole of which we are con- 
sidering parts), and let names which have nothing in com- 
mon, but which between them contain the whole idea under 
consideration, be called contraries in, or with respect to, that 
universe. Thus, the universe being'mankind, Briton and alien 
are contraries, as are soldier and civilian, male and female, etc.; 
the universe being animal, man and brute are contraries, etc." 

(lb. pp. 37, 38.) 

1130. "One inflexion, or one additional word, may serve to 
signify a contrary of any kind: thus, not man is effective to 
denote all that is other than man." 

(lb. p. 39.) 



632 APPENDIX. [ §§ 1133, 1133, 

1131. "Every negative proposition is affirmative and every 
affirmative is negative. Whatever completely does one of the 
two, include or exclude, also does the other. If I say that 'No 
A is B,' then b being the name of everything not B in the uni- 
verse of the proposition, I say that 'Every A is b ;' and if I say 
that 'Every A is B,' I say that 'No A is b.' Whether a lan- 
guage will happen to possess the name B or b, or both, depends 
on circumstances of which logical preference is never one, 
except in treatises of science. The English may possess a term 
for B, the French only for b : so that the same idea must be pre- 
sented in an affirmative form to an Englishman, as in 'Every A 
is B,' and in a negative one to a Frenchman, as 'No A is b.' 

From all this it follows that it is an accident of language 
whether a proposition is universal or particular, positive or 
negative. 

We having the names A and B, may be able to say, 'Every A 
is B': another language which only names the contrary of B 
must say, 'No A is b.' A third language in which A's have not 
a separate name, but are only individuals of the class C, must 
say, 'Some C's are B's; while a fourth, which is in the further 
predicament of naming only b, must have it, 'Some C's are 
not b's/ " 

(lb. p. 40.) 

1132. "In all assertions, however, it is to be noted, once for 
all, that formal logic, the object of this treatise, deals with 
names and not with either ideas or things to which these names 
belong." 

(lb. p. 42.) 

1133. "Thus A and B divested of all specific meaning, are 
really names as names, independently of things: or at least 
may be so considered. For the truth of the proposition, under 
all meanings, gives us a right to suppose, if we like, that names 
are the meanings, that is to say, that we may put it thus: 
'When the name A is, the name B is: but the name B is not; 
therefore the name A is not.' 

It is not, therefore, the object of logic to determine whether 
conclusions be true or false; but whether what are asserted to 
be conclusions are conclusions. By a conclusion is meant that 
which is and must be shut in with certain other preceding 
things put in first. It is that which must have been put into a 
sentence, because certain other things were put in. To infer a 
conclusion is to bring in, as it were, the direct statement of that 
which has been virtually stated already — has been shut in. 

Wlien we say 'A is B, B is C,' we conclude 'A is C,' it would 
be more correct to say, 'A is B, B is C,' we have concluded 'A 
is C " 

(lb. p. 43.) 



§§ 1134, 1135.] CONCLUSIONS. 633 

1134. "Inference does not give us more than there was 
before: but it may make us see more than we saw before: 
ideally speaking, then, it does give us (in the mind) more than 
there was before. But the homely truth that no more can 
come out than was in, though accepted as to all material objects, 
even by metaphysicians — who are generally well pleased to 
find the key of a box which contains what they want, though 
sure that it will put in no more than was there already, has been 
applied to logic, and even to mathematics, in depreciation of 
their rank as branches of knowledge. Those who have made 
this strangest of human errors must have assumed an ideal 
omniscience and looked at human imperfection objectively. 
Omniscience need neither compare ideas nor draw inferences: 
the conclusion which we deduce from premises is always pres- 
ent with them; truths are concomitants, not consequences. 
When we say that one assertion follows from another, we 
speak purely ideally, and describe an imperfection of our own 
minds: it is not that the consequence follows from the prem- 
ises, but that our perception of the consequence follows our 
perception of the premises: the consequence, objectively speak- 
ing is in and with, and of, the premises. We speak wrongly if 
we speak ideally, when we say that 'A is C,' is in 'A is B and B 
is'C: in fact, it is only by giving an objective view to the argu- 
ment that we can even assert that it will be seen. To unculti- 
vated minds, this simple conclusion is never concomitant with 
the premises, and only with some difficulty a consequence. 
From the certainty that a consequence may be made to come 
out, which is an allegorical use of the word out, we assume a 
right to declare by the same sort of allegory that it was in. 
The premises, therefore, contain the conclusion: and hence, 
some have spoken, as if in studying how to draw the conclu- 
sion, we were studying to know what we knew before. All the 
propositions of pure geometry, which multiply so fast that it is 
only a small and isolated class, even among mathematicians, 
who know all that has been done in that science, are certainly 
contained in, that is necessarily deducible from, a very few 
simple notions. But to be known from these premises is very 
different from being known with them." 

(lb. pp. 44, 45.) 

1135. "The study of logic, then, considered relatively to 
human knowledge, stands in as low a place as that of the 
humble rules of arithmetic, with reference to the vast extent of 
mathematics and their physical applications. Neither is the 
less important for its lowliness : but it is not everyone who can 
see that. Writers on the subject frequently take a scope 
which entitles them to claim for logic one of the highest places : 



634 APPENDIX. [ §§ 1136-1140. 

they do not confine themselves to the connection of premises 
and conclusion, but enter upon the periculum et commodum of 
the formation of the premises themselves. In the hands of Mr. 
Mill, for example, (and to some extent in those of Dr. Whately), 
logic is the science of distinguishing truth from falsehood, so 
as both to judge the premises and draw the conclusion, to com- 
pare name with name, not only as to identity or difference, but 
in all the varied associations of thought which arise out of this 
comparison.' 7 

(lb. p. 46.) 

1136. "But is in the sense 'is equal to,' does satisfy all the 
conditions. This sense of is, namely, agreement in magnitude, 
is the copula of the mathematician's syllogism when he is rea- 
soning on quantity only." 

(lb. p. 52.) 

1137. "There are common uses of the word which are not 
admitted in logic : and, among them, one of the most common, 
connection of an object with its quality and of an idea with one 
of its constituent or associated ideas. 

As when we say, k The rose is red,' Prudence is desirable,' 
here the logical conditions are not satisfied. For example, 
'Ked is the rose,' though a poetical inversion of the first asser- 
tion, is not logically true. It is usual to consider such proposi- 
tions in logic as elliptical ; thus, 'The rose is red,' is considered 
as 'The rose is a red object, or an object of red color;' in which 
the is now takes one of the senses which allows of conversion." 

(lb. p. 52.) 

1138. "The is of agreement in particulars may always be 
reduced to the is of identity by alteration of the predicate; 
thus, 'Every A is B in color,' is 'Every A is a thing having the 
color of one of the B's.' " 

(lb. p. 53.) 

1139. "Thus, when we say, "All animals require air,' or that 
the name requiring air belongs to everything to which the name 
animal belongs, we should understand that we are speaking of 
things on this earth : the planets, etc., of which we know noth- 
ing, not being included." 

(lb. p. 55.) 

1140. "Contrary names, with reference to any one universe, 
are those which cannot both apply at once, but one or the other 
of which always applies. Thus the universe being man, Briton, 
and alien are contraries ; the universe being property, real and 
persona] are contraries. Names which are contraries in one 
universe are not necessarily so in a larger one. Thus, in geom- 
etry, when the universe is one plane, pairs of straight lines are 
either parallels or intersectors, and never both: parallels and 



§§1141-1143.] SOME. 635 

intersectors are then contraries. But when the student conies 
to solid geometry, in which all space is the universe, there are 
lines which are neither parallels nor intersectors; and these 
words are then not contraries. But names which are con- 
traries in the larger and containing universe are necessarily 
contraries in the smaller and contained, unless the smaller uni- 
verse absolutely exclude one name, and then the other name is 
the universe." 

(lb. p. 55.) 

1141. "It has been proposed to consider the universal prop- 
ositions as definite with respect to quantity; but this is not 
correct. The phrase 'All X's are Y's,' does not tell us how 
many X's there are, but that, be the unknown number of X's in 
existence what it may, the unknown number mentioned in the 
proposition is the same. That which is definite is the ratio of 
the number of X's of the proposition to the X's of the universe. 
So understood, however, the 'definite quantity,' as an abbrevi- 
ation, may be said to belong to universals. And the indefinite- 
ness of the particular proposition is only hypothetical. It is in 
our power to suppose the sum to be one-half of the whole, or 
two-thirds, or any other fraction. 

The quantity of the subject is expressed; that of the predi- 
cate, though not expressed, is necessarily implied by the mean- 
ing of the language. The predicate of an affirmative is par- 
ticular; the predicate of a negative is universal. If I say 'X's 
are Y's,' even though I speak of all the X's, I only really speak 
of so many Y's as are compared with X's and found to agree; 
and these need not be all the Y's. 'Every horse is an animal' 
declares that so many horses as there are to speak of, so many 
animals are spoken of : and leaves it wholly unsettled whether 
there be or be not more animals left. But if I should say 'X's 
are not Y's,' though it should be only one X, as in 'this X is not 
a Y,' yet I speak of every Y which exists. The assertion is, 
'this X is not any one whatsoever of all the Y's in existence.' *■ 

(lb. p. 57.) 

1142. " 'Some' usually means a rather small fraction of the 
whole; a larger fraction would be expressed by 'a good many;' 
and somewhat more than half by 'most;' while a still larger 
proportion would be 'a great majority,' or 'nearly all.' A per- 
fectly definite particular, as to quantity, would express how 
many X's are in existence, how many Y's, and how many of the 
X's are or are not Y's : as in '70 out of the 100 X's are among the 
200 Y's.' " 

(lb. p. 5&) 

1143. "Again the word negative had better be viewed as not 



636 APPENDIX. [ §§ 1144-1147. 

so much presenting exclusion for its first idea, as inclusion in 
the contrary." 

(lb. p. 68.) 

1144. "The other view which I here propose is really a differ- 
ent mode of looking at that just given. By the time we have 
made every name carry its contrary, as a matter of course, we 
become prepared to take the following view of the nature of a 
proposition. 

A name by itself is a sound or a symbol : its relation to things 
(be they objects or ideas) is twofold. There may be in rerum 
natura, that to which the name applies, or there may not. 1 
do not here speak of how many things there may be to which a 
name applies : it is not essential to know whether they be more 
or fewer, either absolutely or relatively. The introduction of 
contraries may be made the expulsion of quantity. With refer- 
ence to application then, let a name be called possible or impos- 
sible, according as the thing to which it applies can be found or 
not. A name may be compounded of others; the compound 
name being that of everything to which all the components 
apply. Thus, wild animal is the name of all things to which 
both the names of wild and animal apply. To call this com- 
pound name impossible, is to say that there is not such a thing 
as a wild animal : to call it possible is to say that there is such 
a thing." 

(lb. pp. 105, 106.) 

1145. "The affirmative proposition requires the existence 
of both terms." 

(lb. p. 111.) 

1146. "Thus, P, Q, R, being certain names, if we wish to 
give a name to everything which is all three, we may join them 
thus, PQR." 

(lb. p. 115.) 

1147. "The other exclusion may involve, on the same terms, 
an error of the same kind ; or may equally be the expression of 
arbitrary will : but there is what is more reasonably matter of 
opinion about it. 

Aristotle will have no contrary terms: not-man, he says, is 
not the name of anything. He afterwards calls it an indefinite 
or aorist name, because, as he asserts, it is both the name of 
existing and non-existing things. If he had here made the 
distinction between ideal and objective, he would have seen 
that man and not-man equally belong to both (objectively) 
existing and non-existing things : man, for example, belongs as 
a name to Achilles and the seven champions of Christendom, 
whether they ever existed in objective reality or not: and not- 
man belongs, in either case, to their horses. I think, however, 



§§ 1148-1152. J NAMES. 637 

that the exclusion was probably dictated by the want of a defi- 
nite notion of the extent of the field of argument, which I have 
called the universe of the proposition. Adopt such a definite 
notion, and, as sufficiently shown, there is no more reason to 
attach the mere idea of negation to the contrary, than to the 
direct term." 

(lb. p. 128.) 

1148. "But it can hardly be affirmed that any one admitting 
not-man as a name, should thereupon refuse to recognize the 
identity of 'horse is not man with horse is not-man. ' The mid- 
dle term is to be distributed in one or the other of the premises. 
By distributed is here meant universally spoken of. I 
do not use this term in the present work, because I do not see 
why, in any deducible meaning of the word distributed, it can 
be applied to universal as distinguished from particular. In 
using a name, it seems to me that we always distribute : that 
is, scatter as it were, the general name over the instances to 
which it is to apply. When I say, some horses are animals, J 
distribute certain horses among the animals: and when all, 
all." 

(lb. p. 137.) 

1149. "When contraries are introduced, the distinction 
between positive and negative is made to appear what it really 
is, one of language, or rather one of choice of names. But the 
distinction of form is not abolished, but is exactly what it was 
before." 

(lb. p. 139.) 

1150. "The distinction may be easily illustrated by example. 
'All the planets but one/ is a particular proposition; it is 'some 
planets,' there is no one planet of right included in it, but 'all 
the planets except Neptune' is a universal proposition : 'a-planet- 
not-Neptune' is a name of Mercury, of Venus, etc. ; and of every 
planet it can be stated whether it be in the name or not. That 
which is true inferentially of 'all the planets but one' left par- 
ticular, is true of all the planets but Neptune; but that which 
is true of the latter, is not necessarilv true of the former." 

(lb. p. 143.) 

1151. "Thus, man and rational animal are not identical 
names, qua names, for they neither spell nor sound alike: the 
identity understood is that of meaning; where one applies, 
there shall the other applv also." 

(lb. pp. 146-147.) 

1152. "All the theory of names, their application or non- 
application, may be applied to propositions, their truth or false- 
hood. To say that a proposition is true in a certain case, is to 
say that a certain name applies to a certain case; to say that it 



638 APPENDIX. [ §§ 1153-1157. 

is false, is to say that a certain name does not apply, but that 
its contrary does. That contrary is what logicians usually call 
contradictory, and the name is not simply true or false, but the 
adjective attached to the proposition. The conditions under 
which we are to speak, limit us to a number of cases which con- 
stitute what we may now call, not the universe of the names in 
the propositions, but the universe of the truth or falsehood of 
the propositions." 

(lb. p. 147.) 

1153. "I am compelled to use the words contrary and con- 
tradictory as synonymous: at which compulsion I am well 
pleased, never having seen any good reason why, in the science 
which considers the relations of dicta, the contraria should be 
anything but the contra dicta." 

(lb. p. 148.) 

1154. "And just as in a universe of names, every name 
introduced is supposed to belong or not to belong, to every 
instance in that universe: so in a universe of propositions, I 
suppose every proposition, or its contrary, to apply (whether 
it be or be not known which applies) in every instance." 

(lb. p. 149.) 

1155. "The question of a premise being right or wrong, in 
fact or principle, unless, indeed, it contradicts itself, does not 
belong to logic: nor could it so belong unless logic were made, 
in the widest sense, that attempt at the attainment of tho 
nitio vert, which some have denned it to be. All that relates to 
the collection of true premises with respect to the vegetable 
world, belongs to botany; with respect to the heavenly bodies, 
to astronomy; with respect to the relation of man to his cre- 
ator, to theologv." 

(lb. p. 239.) 

1156. "As regards knowledge, there must likewise be a 
transition or change; and the act of knowing includes always 
two things. When we consider our mental states as knowl- 
edge, the same law holds. We know heat by a transition from 
cold; light, by passing out of the dark; up, by contrast to down. 
There is no such thing as an absolute knowledge of any one 
property; we could not know "motion," if we were debarred 
from knowing "rest." No one could understand the meaning 
of a straight line without being shown a line not straight, a 
bent or crooked line." 

(Bain's Deductive and Inductive Logic, p. 3.) 

1157. "Our knowledge of a fact is the discrimination of it 
from differing facts, and the agreement or identification of it 
with agreeing facts. The only other element in knowledge is 



§§ 1158-1161.] LOGICIANS. 639 

the retentive power of the mind, or memory, which is implied 
in these two powers." 

(Ib. p. 4.) 

1158. "The Contradictory Opposition of terms is when they 
differ only in respectively wanting and having the particle 
"not," or its equivalent. One or other of such terms is appli- 
cable to every object." 

(Krause's Vocabulary of Philosophy, p. 513.) 

1159. 'Elimination (elimino, to throw out), in Mathematics, 
is the process of causing a function to disappear from an equa- 
tion, the solution of which would be embarrassed by its pres- 
ence there. In other writings the correct signification is, 'the 
extrusion of that which is superfluous or irrelevant." 

(Ib. p. 155.) 

1160. "As all nature is bound together by certain common 
qualities and relations, human knowledge will be found to con- 
sist chiefly, if not wholly, in comparing resemblances, or con- 
trasting differences: and this is done with little trouble, for we 
perceive intuitively the agreement or disagreement of the 
objects presented to us." 

(Brewster's Encyclopedia Art. Logic, p. 125.) 

1161. "It may confidently be asserted that there is no 
department of human speculation and inquiry in which so 
many contradictory opinions are entertained as in the science 
or art of logic. For the last five-and-twenty centuries, system 
has followed system in rapid succession ; and one generation of 
logicians after another have been chiefly occupied in refuting 
or modifying the principles and correcting the misstatements 
of their predecessors. No sooner has a particular logical sys- 
tem obtained a footing in some locality in the republic of letters 
and become incorporated with the general routine of philosoph- 
ical education, than some aspiring and ambitious speculator 
has called in question its fundamental principles, or subjected 
its practical rules to supervision and amendment. 

From Zeno to modern times, every theoretical logician has 
flattered himself in his day, that he had placed logic on a firm 
basis, not to be disturbed as long as the world lasted. 

He has flattered himself with the idea that it was 
his fortunate lot to chase from the science every vestige of 
doubt, to reconcile every real and apparent contradiction and 
to make to all future generations the path of knowledge and 
science indisputably plain and of ready and agreeable access. 

And the same spirit animates the philosophical logician of 
the present hour in every direction where his science is known 
and cultivated. Every speculator has a system of his own 
with which strangers do not intermeddle. He is the sole 



640 APPENDIX. [ §§ 1162-1164. 

champion of his own theory and the herald of his own fame. 
He, too, labors under the cheering anticipation that he is put- 
ting the finishing stroke to the science, and silencing forever, 
throughout the philosophical world, the voice of doubt and 
contention." 

(Blakey's Hist. Sketch of Logic, p. 16.) 

1162. "Though he may have all the learning of the East, 
and all the talent of Christendom centered in his own person, 
yet he knows full well that apart from his own professional 
chair or private study, he will not find a single cultivator of the 
same science entirely agreeing with him, either on the funda- 
mental principles of logical philosophy, or on the best modes of 
applying them. 

But this does not discourage him or ruffle the equable current 
of his self-complacency. He has the advantage over those who 
have gone before him, hoping unto death the same thing as 
himself; inasmuch as he reasons that if there ever is to be a 
time when the principles of his science are to be known and 
unalterably fixed, he may be the fortunate instrument in this 
grand and noble achievement. While there is life there is 
hope; and this consideration is sufficient to sustain him in his 
labors amidst the mass of disappointment that lies behind 
him." 

(lb. p 16.) 

1163. "The speculative aspects under which logic has 
appeared in different ages and countries have not been more 
checkered and varied than its external fortunes. It has at one 
time revelled in unbounded authority and power, and, yet, at 
another, been doomed to the bitter humiliation of abject serv- 
itude and dependence. It has been the petted child of courts 
and monarchs, and yet been rivalled by the beggar in the street. 
It was once the art of arts, the science of sciences, and the 
proudest emblem in the escutcheon of the philosopher. The 
warrior ventured not to battle with it, nor could the lawyer on 
the bench, or the theologian in the pulpit acquit himself with 
grace, unless versed in its canons and rules. Notwithstanding 
however, all this power and grandeur, we have witnessed the 
science scouted from many influential universities, and, where 
admitted, it was only on the condition of becoming a humble 
menial and a willing slave." 

(lb. p. 17.) 

1164. "In spite, however, of all such reverses logic has 
within it a vigorous principle of vitality. Like the Phoenix, 
it is continually rising from its own ashes. It never allows 
mankind to wander far nor long without pressing its claims 
and obtruding its counsels and admonitions upon them. It 
must, therefore, have a permanent hold on our sympathies, 



§§ 1165-1167.] NATURE OF MIND. 641 

some fixed root in our nature, or it would have been obliterated 
long ago from the book of knowledge. Astrology and alchemy 
never tantalized human reason so severely, for, what can pre- 
sent a greater anomaly to the understanding than that 
logic calling itself a science; having chairs in universi- 
ties set apart for its special cultivation; witnessing its 
professors taking the first rank among the acute and profound 
of our race, and pointing with exulting pride to more than a 
thousand distinct treatises on the subject which have emana- 
ted from their pens within the last three hundred years ; that 
logic, we say, should under these circumstances not be able 
to furnish two logicians of any country who can agree in any 
one common principle of this science, nor be able to state to 
what particular or general uses it can be applied, must present 
to the candid mind one of the most striking phenomena in the 
entire range of human thought. Can any subject in the whole 
circle of the sciences present such a lack of unanimity or a more 
cheerless and desponding aspect? The use of the word logic 
is almost the only thing which disputants have in common. If 
we venture a step beyond this and ask for a definition of what 
is implied in it, we are instantly stunned with a thousand 
discordant voices from all parts of the world." 

(lb. p. 18.) 

1165. This work is not a book on psychology or meta- 
physics, and, although certain writers on logic have had consid- 
erable to say on those subjects, I have avoided them as far as 
possible. And yet the use of the Keasoning Frame has led me 
to adopt the views of Mr. Prince in his work on "The Nature of 
Mind, "and several of Mr. Haig's views which are contained in 
his work on "Symbolism." 

I make the following extracts from those works: 

1166. "When men have once acquiesced in untrue opinions," 
remarks Hobbes, "and registered them as authenticated 
records in their minds, it is no less impossible to speak intelli- 
gibly to such persons than to write legibly on a piece of paper 
already scribbbled over." 

(Prince's Nature of Mind, p. 3.) 

1167. "According to the Theory of Aspects, consciousness 
and nerve motions (vibrations,) are only different aspects of one 
and the same underlying substance which is unknown. This 
view has perhaps been as clearly expressed by Prof. Bain, as 
by anyone else, when he says, 'The one substance, with two sets 
of properties, two sides, (the physical and the mental,) a double- 
faced unitv, would seem to comply with all the exigencies of 
the case.' " 

(lb. p. 14.) 

41 



642 APPENDIX. [§§1168-1172. 

1168. "The same notion has thus been described by Lewes : 
'There may be every ground for concluding that a logical pro- 
cess has its correlative physical process and that the two 
processes are merely two aspects of one event.' " 

(lb. p. 14.) 

1169. And again: "The two processes are equivalent and 
the difference arises from the difference in the mode of appre- 
hension." 

(lb. p. 14.) 

1170. "Thus, Mr. Spencer, who, as a psychologist, has treated 
the matter in a masterly manner, maintains this view of dif- 
ferent aspects. 'For what,' he says, 'is objectively a change in a 
superior nerve-center, is subjectively a feeling, and the dura- 
tion under the one aspect measures the duration of it under the 
other.' And the same thing is repeated in other passages. 
But this is no explanation, as Mr. Spencer, himself, tacitly 
recognizes when he later adds, 'though accumulated observa- 
tions and experiments have led us by a very indirect series 
of inferences to the belief that mind and nervous action are the 
subjective and objective faces of the same thing, we remain 
utterly incapable of seeing and even of imagining how the two 
are related." 

(lb. p. 17.) 

1171. "We can have no consciousness without a material 
substance, the brain, nor without the activity of the brain." 

(lb. p. 45.) 

1172. "Now, that matter, of which conscioussness is the 
reality, must be subject to the laws which govern matter. One 
of these laws is the law of inertia. According to this, matter 
cannot of itself change its own state. Matter at rest, must 
forever remain at rest, unless something outside of itself dis- 
turbs it and puts it in motion. Matter in motion must forever 
persist in motion till something outside of it checks it. Matter 
exhibited under one property, must forever be exhibited under 
that property, unless some external force causes it to be exhibi- 
ted under another. Whatever be the state of matter at a given 
moment, it must always remain in that state until outside 
agencies effect a change. This is a universal law, it has no 
exception. To this law then, the 'matter of the mind' must be 
subject. 

Let us apply it and see what it means. It means this: that 
no change of any kind, chemical or physical, can occur in the 
protoplasm of the brain, without the interference of outside 
agencies; that no vibration or pulsation can occur among the 
protoplasmic molecules of any cell, unless some cause external 



§§1173,1174.] CONSCIOUSNESS. 643 

to that cell acts upon them; that for the undulations of the 
molecules — of which consciousness is the reality — to occur, 
some external force is requisite to start them into activity; in 
other words, for consciousness to be present, it is necessary 
that each cell should be stimulated by something external to 
that cell. The activity of the molecules of no cell can appear 
spontaneously, and, hence, neither can the reality of that activ- 
ity, or consciousness. Consciousness, then, is passive, not 
active; it is conditioned existence, not unconditioned; it is a 
link in a series of events." 

(lb. pp. 94-95.) 

1173. "Such is the inevitable result to which our reasoning 
leads us. If consciousness depends on matter being dis- 
turbed, it must be passive. This is a logical consequence of 
our premises, from which there is no escape. But if our 
thoughts are passive, — if they are merely the molecular dis- 
turbances in themselves, and cannot arise spontaneously, — it 
must bp that the stimulus required for their production, cannot 
be applied in any definite manner at haphazard, but only 
through the anatomical mechanism of the brain, — only through 
the nerve conductors developed for the purpose. The channels 
by which stimuli from without reach the cells of the brain, are 
the centripetal nerves; and any succession of ideas can only 
occur by reason of the neural 'currents,' wherever originated, 
being reflected from one cell to another, along the anatomical 
connections which join the cells; and any objective expression 
of an idea can only take place by reason of the current passing 
again from the brain to the organs of expression, which are the 
muscles. In other words, under normal conditions every mus- 
cular action, every idea, sensation or emotion, requires for its 
production some stimulus originating outside of its own nerv- 
ous center, — that is, it is reflex." 

(lb. pp. 95-96.) 

1174. "To this reflex view there are logical consequences 
from which I see no escape. From a theory that a mental pro- 
cess is the reality of the reflex physiological process to the 
doctrine. of automatism, is a step which we are compelled by 
the force of logical necessity to take, or rather, the two doc- 
trines are essentially the same. For any doctrine which 
removes our thoughts from the control of a hypothetical agent, 
which is independent of external influences, and confines them 
to certain channels in which they are propelled, directly or in- 
directly, by stimuli (external or internal,) is practically auto- 
matism. 

Under the reflex view, spontaneity, in the sense that any idea 
or state of mind cau arise, except as the resultant of some other 



644 APPENDIX. [§§1175-1179. 

idea by which it is conditioned, is impossible. Reflex is, con- 
sequently, equivalent to automatic." 

(lb. p. 100.) 

1175. "When it is said that mental processes are automatic, 
I do not conceive that it is necessarily meant that we are identi- 
cal with or like machines in every particular. 

For instance, human beings grow and generate other human 
beings, functions not possesssed by machines. When it is said 
that we are automata, or, that our mental processes are auto- 
matic, I understand that all that is meant is that our thoughts, 
sensations, volitions and actions, follow in certain grooves 
or channels which have their analogies and equivalents in the 
anatomical mechanism of the brain, and that the presence of 
every state of mind is conditioned by the anatomical structure 
and physiological working of the brain. Automatism is then 
synonymous with reflex action." 

(lb. pp. 105-106.) 

1176. "There is one thing which must not be overlooked, 
and this is, that whatever powers of self-determination we may 
have, every action is determined by the strongest motive. How- 
ever we may act, we cannot act contrary to the strongest 
motive; for the moment we conclude to act in opposition to 
what was the strongest motive, the new motive, whatever it 
be, if it be only the desire to show that we have the power to do 
so, becomes the strongest motive, overwhelming the preceding, 
and determining action. Whatever motive determines action 
is the strongest, — else it would not so determine us, — and we 
are compelled to act according to it." 

(lb. p. 141.) 

1177. "The gentlest and most practical of all the Apostles 
of Christ, left this deep truth: 

"If any man offend not in word, the same is a perfect man, 
and able to bridle the whole body;" and his Master, himself, 
said, "For every idle word that men do speak they shall give 
account in the day of judgment; for by thy words thou shalt 
be condemned." 

(Haig's Symbolism, p. 11.) 

1178. "Thinking is internal reasoning, or, reasoning to our- 
selves. Reasoning is external thinking, or thinking expressed 
in signs, words, symbols, intelligible to others. The one is 
private and peculiar to the individual man; the other is the 
same thing when made common to all mankind possessed 
of language and sufficient intelligence to comprehend it." 

(lb. p. 1.) 

1179. "But these conventional words, which are called the 
names of the external things, are themselves the only things 
about which we can reason or hold any discussion, or have 



§§ 1180-1185.] SYMBOLISM. 645 

any question in truth; because they are the only things or 
objects which men have, or can have, in common. Words are 
the only common objects which men possess jointly, or can 
compare together, in any possible question or discussion that 
can be raised betweeen man and man. Man's mind must start 
with a symbol or symbols." 

(lb. p. 3.) 

1180. "Our reasoning must begin and end with words — our 
reason has no other mutual instrument and no other mutual 
object; and though the faith of every man is quite fixed, and as 
certain to himself as the rock on which it stands, that lan- 
guage is not all that exists in the universe, yet it is all that 
exists in human cognition — it is all that men can compare 
together in every question and every discussion, and it forms 
every possible conclusion at which men can jointly arrive by 
their most earnest and careful thinking and reasoning." 

(lb. p. 4.) 

1181. "Therefore, in the first place, it is clear that the reas- 
oner always assumes the existence of the thing about 
which he proposes to reason. In the second place, it is equally 
certain that the reasoner cannot propose any question concern- 
ing the thing to be discussed, without he first assumes the pos- 
sibility of such question. Thus, certain Existences and Possi- 
bilities — or, to use their logical names, Categories and Predi- 
cates — are unavoidably necessary, and must always be 
assumed in every discussion, before men can possibly think 
or reason together in any way whatever." 

(lb. pp. 4-5.) 

1182. "Ideas are mental to the individual, but must be 
verbal to more than one man, or to mankind — all ideas arc 
words in the mind." 

(lb. p. 77.) 

1183. "But what the mind holds is not the outer world, but 
the minute undulations, motions and forms which reach the 
brain ; but which motions exist only in the nerves of each man's 
own body caused bv the outer world." 

(lb. p. 107.) 

1184. "Logic is merely the laws by which we properly sub- 
stitute one verbal expression for another, as its equivalent, 
in ordinary language." 

(lb. p. 148.) 

1185. "How and by what means Man is to distinguish true 
symbols — true words from false words — thus becomes, to our 
human intellect, the question of questions; the end and object 
of all purely intellectual thinking and reasoning — the great 
riddle of philosophy — the science of Truth." 

(lb. p. 160.) 



646 APPENDIX. [§§1186-1194. 

1186. "Thus we think of things and speak of thoughts, but 
can only reason to others about words." 

(lb. p. 163.) 

1187. "Men can, therefore, neither speak nor reason of 
either things or thoughts themselves, but only of general 
woids; not because words are everything in nature, but because 
they are everything in general human cognition and human 
reasoning." 

(lb. p. 165.) 

1188. "Words, then, are something; they are signs of 
thoughts: thoughts are something; they are signs of things; 
and as we can know nothing of other men's thoughts of things, 
we can only discuss other men's words for their thoughts of 
things, and we can discover and invent new thoughts of things 
and express them in some new, clear, orderly and intelligible 
words." 

(lb. p. 186.) 

1189. "But how are we to distinguish true axioms from the 
false ones, since false axioms have been assumed as self-evi- 
dent? I know no answer to this except by comparing and 
reflecting upon the logical deductions and relations which we 
can make from them." 

(lb. p. 216.) 

1190. "Logical inference is always hypothetical in form, 
and depends on the premises being true; but the inference 
itself, is absolute truth; the connection and mental deduction 
are absolute and necessary, and the conscientious intellect 
must distinguish between the necessary and the possible." 

(lb. p. 217.) 

1191. "But there is, in fact, and can be, no such real exist- 
ence as a general thing, or general idea apart from the word — 
the general term accepted by mankind in general." 

(lb. p. 226.) 

1192. "In my opinion, the phrenologists have given far the 
best analysis of the human mind, and its natural powers and 
tastes and capacities and sentiments." 

(lb. p. 391.) 

1193. "Of course, for philosophers, or men of science, to 
argue and reason with one another without defining their 
words, is about as reasonable and useful as might be the mut- 
ual confabulations of two men who do not understand each 
other's language, and are blind to each other's signs." 

(lb. p. 398.) 

1194. "However, it is a most idle work to reason with indef- 
inite words. That ought to be contemptible to the man of 
intellect." 

(lb. p. 410.) 



§§ 1195, 1196.] WORDS. 647 

1195. "When men of science speak of certain curves and 
motions and forces and atoms, they are talking of words whose 
scientific meanings are fixed; and they can crucify a falsity 
with logic. But metaphysics and politics and sociology are 
not yet scientific, and therefore, the spirit of falsehood can in 
such pseudo-sciences well contend with the spirit of truth, by 
means of false or ambiguous words." 

(lb. p. 437.) 

1196. "And when we examine the likenesses or resem- 
blances of the words of all languages to each other, we ulti- 
mately find that we can reduce them all to the two classes of 
nouns and verbs, for things and actions." 

(lb. p. 438.) 



INDEX. 

(The numbers refer to pages.) 



Abelard a Nominalist, 611. 

Abscissio infiniti, 335. 

Accident, a predicable, 11. 

Adamson Prof., extract from, 625. 

Additional premises, 78. 

Adjectives, are names, 10; indefi- 
nite, 110. 

A, E, I, O, 303; inferences from, 
397; symbols, 15. 

Affirmation of the consequent, 220, 
258. 

Affirmative, meaning of, 14; do not 
distribute the predicate 372; imply 
existence, 636. 

Affirmo, 15, 303. 

Alexander of Aphrodisias, conver- 
sion of " E," 456. 

Alexander of Halles, a schoolman, 
609. 

All, DeMorgan on, 598; not the op- 
posite of some, 359. 

Alternates, 13. 

Alternatives, are indefinite, 79; meth- 
od of stating, 39; reduction to cat- 
egoricals, 169, 171; repeating the 
subject, 39; rule for combining, 
174; stating of, 181. 

Alternants, 127; expression of, 127; 
only one can be true, 182. 

Ambiguity, 181, 398. 

Animals, reasoning powers, 4. 

Anselm, a schoolman, 609. 

Antisthenes, the first Nominalist, 
607. 

Appendix, 607. 

Apprehension, of abstract ideas, 12. 

Arguments, two disjunctive prem- 
ises, 186. 

Aristotle, on the fourth figure, 362; 
allowed no contrary terms, 636; 
conversion of universal negatives, 
408; false premises, 318; first sci- 
entific logician, 607; on singular 
terms, 58; order of premises, 373; 
puts major premise first, 373; 
sketch of, 608; dictum, 304. 

Aristotelian Sorites, 365. 

Arithmetic and logic, 6; on a par, 
33. 



Automatism is reflex action, 644. 

Bocardo, Sorites in, 366. 

Bacon, Francis, on induction, 90; 
an opponent of the scholastics,6l0. 

Bain, A., contraposition, 409; criti- 
cism of the syllogism, 312; exam- 
ple from, 183; example from, 268; 
271; illicit process, 321; material 
obversion, 426; negative premises, 
323; on dilemmas, 262; on fallacies, 
595; on negation, 374; on negative 
premises, 323; on "or," 205; ob- 
version, 412; on the dilemma, 279; 
propositions are true or false, 372; 
on quantification, 389; rule for 
obversion, 412; singular premises, 
374; syllogism with two singular 
premises, 374; why quantification 
necessary, 389. 

Barbara, 339; Celarent, etc., 361; ex- 
ample in, 252. 

Baroco, 346; example in conversion, 
441; Sorites in, 366. 

Barrow, Dr., on confusion, 7. 

Bergmann, 626. 

Bocardo, 351 ; example in conversion, 
443. 

Boole, example of his method, 180; 
a reformer in logic, 618; on " or," 
205; sign for "some," 60; alge- 
braical system, 56; doctrine of 
thought, 628; formulas, 629; sys- 
tem, criticism on, 383. 

Booleian, an algebraic system, 318. 

Bolzano, diagrams, 56. 

Bossuet, on finding truths, 613. 

Bowen, Infinitation, 426. 

Brain, a thinking machine, 42; sub- 
ject to the law of inertia, 642. 

Bramantip, 353. 

Brewster's Encyclopedia, on the syl- 
logism, 620. 

Brown, Thos., on the syllogism, 617. 

Buridan, a schoolman, 609. 

Campanella, an opponent of the 
scholastics, 610. 

Campbell, G., on syllogisms, 615. 

" Can," meaning of, 411. 

" Cannot," meaning of, 411. 



650 



INDEX. 



Camenes, 354. 

Camestres, 343; example in, 256; ex- 
ample in conversion, 440. 

Capital letters in Mnemonic lines, 
362. 

Casually, meaning of, 96. 

Categorematic, 12. 

Categoricals, meaning of, 22. 

Categories are useless, 608. 

Celarent, 339, 345. 

Certainty, and law of contradiction, 
139. 

Cesare, 342; example in, 257. 

Chrysippus, a follower of Aristotle, 
607. 

Clarke, Father, criticism on the 
Galenian figure, 362; on the 
fourth figure, 362. 

Classification, 9. 

Coincidental propositions, 420. 

Combinations, for three terms, 64; 
inconsistent, reading of, 128; law 
of, 33; made mechanically, 49; 
method of making, 33; of eight 
things, 107; of four terms, 107; 
represent a bundle of propositions, 
138; represent propositions, 85; 
see Law of Combinations; true 
and false depend upon elim- 
inations, 138; uneliminated, 178; 
uneliminated, reading of, 128. 

Complex, propositions, 466; terms, 
127. 

Conceptualism, doctrine of, 609. 

Consciousness is difference, 623. 

Conclusions, are shut in, 632; consis- 
tent with the premises, 138; from 
independent premises, 316; of spe- 
cial value, 61; there are many, 584; 
various methods of expression, 
584; weakened, 359; why so called, 
304. 

Condillac, on mathematical knowl- 
edge, 615; theory of logic, 626. 

Conditionals, Miss Jones on, 223; 
relate to prior circumstances, 
211. 

Conditional propositions, 211. 

Confusion of ideas, example of, 621. 

Conjunctions, no sign for, 37; stating 
of, 119. 

Consistency, demands of, 26. 

Consistents, 375; example of, 473, 
480, 483, 482; meaning of, 375. 

Consistent propositions, how divid- 
ed, 491; what are, 376, 492. 

Contradictories, 128, 375; complete, 
process for, 167; conversion into 
alternatives, 399; conversion of, 
399; eliminate a letter term, 375; 
example of, 467; if one is true, the 



other is false, 371; many can be 
found, 569; method of finding, 
565; of hypotheticals, 224; of prop- 
ositions, 562; what are, 371 ; contra- 
dictory contradictories, 579; of an 
expression, 160. 

Contradiction, law of, 26. 

Contradictoriness, of propositions, 
2; sign of, 397. 

Contraposition, 409; of "A," 457; 
meaning of, 427; two contraposi- 
tives, 427. 

Contrapositive, rule for, 428. 

Contrary, is contradictory, 638; 
names represent everything, 631. 

Contraries, 370; what are, 370. 

Contraries in terms, 13. 

Contraversion, ot "0,"419; meaning 
of, 418. 

Converse, meaning of, 407. 

Conversion, 407; a transposition of 
terms, 371; an example of, 434, 
435; by limitation, 361, 408; by 
negation, 410; criticised by Hamil- 
ton, 616; example, All true patriots, 
444; All true philosophers, 440; 
Every true patriot, 441; Salt is 
chloride of sodium, 437; Some 
slaves are not discontented, 442; 
impure, 415; per accidens, 408, 361; 
new readings, 435, 436; of hypo- 
theticals, 225; of particular affirm- 
atives, 409, 416, 371; of partic- 
ular negatives, 371, 416, 408; 
of universal affirmatives, 371; of 
universal negatives, 371, 408; 
per accidens, 409; rules for, 407; 
simple, 361; what is, 371. 

Consequent, granting of the, 218. 

Cousin, V., on logical methods, 621. 

Convertend, meaning of, 407. 

Copula, 14; a particular, 384; inser- 
tion of, 38; meaning of, 37; par- 
ticular, an impossibility, 384; re- 
lation to time, 373; sign of, 37; 
when not necessary, 38. 

Cross-division, 9. 

Darapti, 347; example in, 253. 

Darii, 340; conversion into, 439. 

Data, must be correct, 6. 

Datisi, 349. 

Debate, this system cuts off, 585. 

Definitions, accidental, 8; alterna- 
tive, 179; of disjunctive terms, 
307; Hobbes' of "name," 9; logi- 
cal, 9; necessity of, 8; nominal and 
real, 8; of terms, rules for, 88; 
qualities of, 8; reading of, 307; 
tautological, 9. 

DeMorgan, A., on " all," 598; on dis- 
junctives, 158; on diagrams, 56; on 



INDEX. 



651 



fallacy, 597; on negative terms, 55; 
contraversion, 426; example from, 
512; extracts from, 629; quoted by 
Keynes, 429. 

Denial, affirms truth of the contra- 
dictory, 398; no sign for, 39; when 
equivalent to affirmation, 398. 

Determinants, 127. 

Diagrams, at first blank, 170; ways 
of making, 64. 

Dichotomy, illustration of, 29; illus- 
tration of, 42; meaning of, 28; 
tables of, 30. 

Dictum de omni et nullo, 304. 

Diderot, D. , reasoning a species of 
sensations, 614. 

Difference, a predicable, 11. 

Dilemmas. 262. 

Dimaris, 355. 

Disagreement intuitively perceived, 
639. 

Disamis, 348. 

Discourse, a series of propositions, 
13. 

Discovery by the Reasoning Frames, 
238, 239; new method for disjunc- 
tives, 167; method of finding con- 
tradictories, 585; method of finding 
equivalents, 556; method of find- 
ing inconsistent propositions, 400; 
method of reading inferences, 376; 
of this method, when, 584; of this 
system, 56. 

Disjunctive, proposition, 13; propo- 
sitions, method of solving, 480. 

Disjunctives, 158; affirmative contra- 
dictories, 581; affirmative equiv- 
alents, 580; negative contradic- 
tories, 581; negative equivalents, 
580; expression of, 127; implica- 
tion of, 180; imply opposition, 158; 
indefinite, 79; in the old logic, 
160; often fallacies, 180; predi- 
cates of, 160; reading of, 127; re- 
duction to categoricals, 192; re- 
duction to hypotheticals, 192; 
separated by or, 22; separated by 
or, 158; subjects of, 160. 

Distributed, and undistributed, arbi- 
trary, 16; doctrine of, 16; subjects 
and predicates, 372; doctrine of, 
372. 

Double epicheiremas, 364. 

Division, logical, 9; physical, 9. 

Eliminated combinations, reading of 
386; use of, 58. 

Elimination, 138; a process for in- 
consistent propositions, 138; exam- 
ple of, 497; law of, 31; means to 
throw out, 639; of a letter, meaning 
of, 397; of negative terms, 449. 



Eliminating combinations, meaning 
of, 138. 

Encyclopedia Britannica, extract 
from, 625; induction, 108. 

Enthymeme, 363. 

Epicheiremas, 363. 

Epicurus, a follower of Aristotle, 
607. 

Episyllogism, 363. 

Equipollent, meaning of, 399. 

Equivalent propositions, example 
of, 452. 

Equivalents, description of an idea 
and its opposite, 100; discovery of 
method, 556; example of, 471, 486, 
493, 494, 468; method of finding, 
543; negative, 375; substitution of, 
137; what are, 491. 

Erigena, a schoolman, 609. 

Euclid, criticism of, DeMorgan, 429. 

Eulerian circles, useless for complex 
propositions, 584; system of, 56. 

Examples; All avaracious men re- 
fuse to give money, 144; All fixed 
stars are self-luminous, 153; All 
heated solids give continuous 
spectra, 152; All horned animals 
ruminate, 305; All human crea- 
tures are, etc., 337; All planets are 
subject to gravity, 141; A science 
which furnishes the mind, etc., 
146; A =B, 42; A=Bor C, 184; 
A is B or C, 185; A = B 
or C = D, 186; A = B or C 
= D, 240; A = B or C = D, 282; 
AorB = CorD, 163; AorB=Cor 
D, 167; A or B = C or D, 242; ab = 
cd, 165; A is not true, or B is true, 
187; Ceesar was an usurper, 209; 
Chloride of sodium, 45; containing 
three terms, 140; Cornishmen are 
Englishmen, 311; C is D or A is 
b, 188; Death is not life, 51; 
Double Epicheirema, 364; " Ex- 
cept a man be born of water," etc. , 
533; Every blood vessel is, etc., 
190; Either the witness is perjured, 
183; Either B or C exists, 182; 
finding contradictories to disjunc- 
tives, 569; Fishing party problem, 
504; for practice, 84; from De 
Morgan, 512; from Prof. Jevons, 
510, 511; Gems are either rare 
stones, etc., 114; Granite is not a 
sedimentary rock, 140; Grover 
Cleveland is president of the XL 
S„ 82; " He that is of God heareth 
my words," 142; Iron is metal, 60: 
If A is true, etc., 248; If A = B, 
C = D, 249; If A = B, C = D and 
E — F, 278; If a King of Spain, 



652 



INDEX. 



etc., 281; If iEschines joined, etc., 
272; If a classical education, etc., 
269; If a righteous God, etc., 
248; If a straight line, etc., 226; If 
Caesar was an usurper, 210; If 
force is expended, etc., 219; If 
honesty is not the best policy, 
447; If patience is a virtue, etc., 
247; If school-masters can claim, 
etc., 273; If science furnishes, etc., 
273; If the barometer falls, etc., 
262; If the education of certain 
children, etc., 216; If the first 
preachers, etc., 215; If the orbit of 
a comet, etc., 268; If the weather 
continues fine, etc., 217; If this 
man were wise, etc., 274; If this 
penis not cross-nibbed, etc., 236; 
If this pen is not cross-nibbed, etc. , 
245; If this river has tides, etc., 220; 
If this river has tides, etc., 221; 
If water is salt, etc. , 250; in finding 
contradictories, 562, 566; in find- 
ing equivalents, 558; in numerical 
reasoning, 462; John is- a man, 
143; Mount Blanc is the highest 
mountain in Europe, 147; Neptune 
is a planet, 150; Never when C is 
D, 254; Not-death is life, 53; Not- 
life is death, 54; No gods are men, 
etc., 358; No savages have the use 
of metals, 308; of a fallacy, 593; 
of a pretended syllogism, 320; of 
a negative equivalent, 378; of a 
simple destructive dilemma, 275; 
of a Sorites, 367; of an in- 
formal syllogism, 368; of cate- 
gorical equivalents, 548, 550; 
553, 554; of equivalent proposi- 
tions, 451; of exclusive proposi- 
tions, 396; of first figure, 330; of 
fourth figure, 332; of Illicit 
Process of the Major, 323; of 
Illicit Process of the Minor, 322; 
of inconsistent propositions, 400; 
of negative equivalents, 551; of 
ob version, 424; of second figure, 
331; of third figure, 331; of the 
third figure, 336; Red colored 
metal is, etc., 176; Salt is chloride 
of sodium, 143, 31, 157; Sodium is 
a metal, 149; Solids or liquids or 
gases, etc., 160; Some dogs are 
all pugs, 436; Some Europeans are 
Englishmen, 310; Some men are 
kings, 320; Some metals are of 
less density than water, 154; The 
case of Caesar, 208; The common 
council problem, 506; The four 
hunters' problem, 505; The mem- 
bers of a board, etc., 179; The 



powers delegated to the U. S. , etc. , 
97; The Queen of England, etc., 80; 
The substance of least density is 
hydrogen, 77; The Tenth Amend- 
ment, 508; The yacht problem, 501. 
The young ladies problem, 503; 
This pen is either cross-nibbed, 
etc., 230, 234; This pen is neither 
cross-nibbed, etc., 235; Those 
poems of Ossian, etc., 335; Wash- 
ington is the capital of the U. S., 
314; Wealth is what is transfera- 
ble, etc., 171; Whales are not true 
fish, 151; What is not chloride of 
sodium is not salt, 48; in induc- 
tion, 535; in reading, 286-300; in 
stating propositions, 284; Life is 
not death, etc., 52; of contradic- 
tory propositions, 423; of Enthy- 
nemes, 363; of equivalent proposi- 
tions, 422; of fallacies, 590; of in- 
ference, 377; of ob version, 425; of 
retroversion, 419; to be worked, 2; 
Murder problem, 500; What is an 
equivalent for A or aB? 136; What 
is an equivalent for AC or AD or 
BD? 136; What is an equivalent 
for A or BC? 135; What is an 
equivalent for AB or AC? 134; 
What is the opposite of A or B? 
129; What is the opposite of A or 
BC? 130; What is the opposite of 
ABC or ABD? 131; What is the 
opposite of a or b or cd? 132; 
What is the opposite of ab? 130; 
What is the opposite of a or b? 
129; What is the opposite of ab or 
ac? 131 ; What is not the substance 
of least density, 78; Whenever C 
= D, etc., 252. 

" Except," a substitute for not, 375; 
translation of , 19. 

Excluding, a substitute for not, 375. 

Excluded middle, law of, 27. 

Exclusive, figure, 335; meaning of, 22: 
205; propositions, 396. 

Exercises for practice, 118, 137, 156, 
206, 260, 282, 603. 

Existence, in Universe of Discourse, 
459; logical, 458. 

Extension, applied to names, 21. 

Facts, this system applied to, 510. 

Fallacies, 2; dilemmas usually are, 
279; examples of, 586; of rela- 
tivity, 596; Argumentum ad 
Hominem, 589; begging the ques- 
tion, 588; confusion of ideas, 589; 
suppression of truth, 590; equivo- 
cation, 587; example of, 593; 
irrelevant conclusion, 589; the im- 
perfect dilemma, 600; irrelevant 



INDEX. 



653 



questions, 596; motion, 600; reason- 
ing in a circle, 588; self contradic- 
tion, 589; Prof. Bain on, 595; Prof. 
De Morgan on, 597; putting more 
questions than one as one, 597. 

False premises, 318. 

Fatalism, Prof. Venn on, 596. 

Felapton, 350. 

Ferio, 341; example in conversion, 
442. 

Ferison, 352. 

Fesapo, 356. 

Festino, 345. 

Few, exclusion of, 63. 

Field of thought (see Universe of 
Discourse). 

Files, subdivisions of sections, 41. 

Figure, each admits six moods, 337; 
Four, Aristotle on, 362; Four, 
Clark on, 362; Four, Galen, 362; 
Fourth, example of, 332; Four, 
Thompson on, 362; Third, exam- 
ple of, 331; Third, example of, 336; 
Two, example of, 331; Two, rules 
for, 359; Two, the exclusive figure, 
335. 

Figures, the, 330; combinations of, 
334; rules of, 333; table of, 333. 

Finding contradictories, 565. 

First figure, 302; example, 330. 

Fresison, 358; example of, 358. 

Form, logical, 17. 

Fourth figure, 303; example of, 332; 
not recognized by Aristotle, 362. 

Fowler, permutations, 424; on "or," 
205. 

Galen, figure four, 362. 

Galenian figure invented by Galen, 
862. 

Gassendi, an opponent of the schol- 
astics, 610. 

Geometrical problems, 227. 

Generalization, 20. 

Genus, a predicable, 11. 

Gerson, J., a schoolman, 609. 

Gloclenian Sorites, 365. 

Grammar and logic, 7. 

Grassman's theory of logic, 629. 

Gregory IX, Bull against Lully, 611. 

Haig's Symbolism, 644. 

Hamilton, Sir Wm., eight forms of 
propositions, 388; his position 
correct, 389; on concepts, 11; on 
denotation and connotation, 21; 
on disjunctives, 204; on "or," 
205; on quantification of predi- 
cate, 616; on reasoning, 3; on 
quantification, 379; rule for quan- 
tification, 379; rule for quantifica- 
tion, 389. 

Hamilton's rule, 396. 



Hegel, G. W. F., on relativity, 615. 

Helvetius, on identical propositions, 
614. 

" He that believeth," etc., 531. 

Historical notes, 607. 

Hobbes, doctrine of thought, 628; 
predicate and subject identical, 
372; reason is nothing but reckon- 
ing, 612. 

Hypotheticals, are indefinite, 208; 
can be converted, 208; considered 
to be universals, 223; contradictor- 
ies of, 224; conversion of, 225; ex- 
amples in conversion, 447; have 
suppressed premises, 208; imply 
permanent relations, 211; indicate 
doubt, 208; meaning of, 22; qual- 
ity of, 224; value of, 222; with in- 
consistent antecedents, 238; with 
inconsistent consequents, 239. 

Idea and opposite, descriptions of, 5. 

Ideas, are words in the mind, 645; 
represent things, 4. 

Identity, is that of meaning, 637; 
law of, 25; see Law of Identity. 

Identifying propositions, 247. 

Identical propositions, 407. 

Illicit process, 321. 

Ignoratio Elenchi, fallacy of, 589; 

Illicit process, of the major , ex- 
ample of, 323; of the minor, ex- 
ample of, 322. 

Imperfect dilemma, fallacy of, 600. 

Inconsistent contradictories, 577. 

Inconsistents, 375. 

Inconsistency, 397; absence of a 
letter, 82; detection of, 57; in 
premises, 82. 

Inconsistents, meaning of, 375; 
method of finding, 406; opposites, 
13. 

Inconsistent propositions, example 
of, 82; propositions, how divided, 
491. 

Independent, premises combined, 
314; propositions, 422; proposi- 
tions, what are, 377. 

Index Expurgatorius, writings of 
Lully, 611. 

Induction, 85; and deduction no 
difference, 98; easy and simple, 2; 
Encyclopedia Brittanica. 108; 
Francis Bacon on, 90; E. E. C. 
Jones on, 108 ; example of, 97; 
gives us all the premises, 99; J. 
S. Mill on, 90; Jevons, J. S., on, 
90, 106; John Locke on, 90; Key- 
nes, J. N., on, 107; method of, 86, 
88; Principles of Science on, 106; 
problem of, 85 ; tentative meth- 
od, 94. 



654 



INDEX. 



Inductive examples, 535; examples, 
method for, 538; problems, solu- 
tion of. 92, 93. 

Infallible logic, 24. 

Inference, 35, 228; an experiment in, 
35; by qualifying, 36; example of, 
469, 475, 484, 485, 488, 489, 492, 
498; example of, 490; from nega- 
tive propositions, 46; from the 
denial of the antecedent, 212; 
immediate, 35; is a result of, 35; 
is hypothetical, 646; mediate, 35; 
what is, 492. 

Inferences, 376; from A E I O, 
397; what are, 376. 

Inferend, 376; meaning of, 491; what 
is, 492. 

Inferring negatives, 413. 

Informal syllogisms, 360. 

Intellectual thought, divisions of, 4. 

Interpretation, this system applied 
to, 510. 

Intension, applied to names, 21. 

Introversion, meaning of, 418. 

Inverse and obverted inverse, 430. 

Inversion, meaning of, 419. 

Irrelevant questions, fallacy of, 
596. 

"Is", affirms existence of names, 
370; definition of, 13; joins sub- 
ject and predicate, 370; no rela- 
tion to time, 373; not equational, 
373. 

Jevons, J. S., an example of a dis- 
junctive, 160; combinations of 
eight things, 107; criticized by 
Keynes, 429; criticized by Miss 
Jones, 326; criticized by Miss 
Jones, 328; doctrine of thought, 
628; Elementary Lessons, 108; ex- 
ample from, 77, 141, 171, 174, 176, 
374; example of negative proposi- 
tions, 374; immediate inference by 
privative conception, 426; incor- 
rect examples, 115; meaning of 
"or," 171; negative propositions, 
374; no question about existence, 
459; on differentiation, 91; on 
equivalent propositions, 110; on 
disjunctives, 158; on induction, 
90; on induction, 106; on negative 
propositions, 374; on "or," 159; 
sign for "some," 60. 

John of Salisbury, on Abelard, 611. 

Jones, E. E. C, 460; conversion of 
hypothetical, 446; criticism of 
Jevons, 326; criticism of the syllo- 
gism, 314; criticism of syllogism. 
325: definition of logic, 3; ex- 
ample of quantification, 385; ex- 
ample from, 447; example of 



quantification, 386, 388; new 
terminology for conversion, 417; 
on conditionals, 223;. on induc- 
tion, 108; on "or," 205: on quanti- 
fication, 384, 386, 387; on theory 
of inclusion in a class, 387. 

Jones, Miss, criticism of Jevons, 328. 

Judgment, a proposition, 13; in old 
logic, 12; is naming, 12. 

Judgments, reciprocal, 415. 

Jurisprudence and logic, 7. 

Kant, Emanuel, divisions of prop- 
ositions, 21; example from, 382; 
on diagrams, 56; on " or," 205. 

Kant's autonomy, a fallacy, 596. 

Keynes, J. N., an apparent illicit 
process,' 443; A Y I, 391; classes 
of propositions, 118; complemen- 
tary propositions, 424; contraposi- 
tion, 427; contraposition of "A," 
457; conversion per accidens 409; 
conversion of "E," 456; criticism 
of Jevons, 429; criticism of 
Thompson, 393; definition of logic, 
3; example from, 184, 185, 245, 
252, 255, 275, 496, 466, 476, 478, 
487, 495, 461; example of enthy- 
memes, 363; example of ob ver- 
sion and conversion, 431; log- 
ical existence, 458; figure One, 
334; figure Two, 335; inversion, 
430; material ob version, 426; ob- 
verted contrapositive, 428; on A 
and n, 394: on A U Y and n, 
394; on A Y I, 391; on contra- 
dictories, 244; on dilemmas, 262; 
on induction, 107; on I U n, 
390; Mr. McColl, 228; on obver- 
sion of complex propositions, 134; 
on ob version, 424; Mr. Wel- 
ton, 229; on "or," 206; on 
predication, 389; on proposi- 
tion n, 392; on propositions 
Y and n, 394; position in re- 
gard to existence, 460; on "some," 
390; on the proposition w, 396 ; 
on the Sorites, 366 ; on quan- 
tification, 380, 389; quotation from 
De Morgan, 429; quotation from 
Thompson, 362 ; rules for con- 
version, 407; rules of the figures, 
333; rules of the Sorites, 365: 
" some," pitfalls of, 390; table of 
conversions, 421; when ob version 
possible, 432. 

Knowledge, absolute, 138; implies 
difference, 24; includes two things, 
638; is discrimination, 638. 

Krause, diagrams, 57. 

Lambert, sign for " some," 60. 

Language, elliptical, 19. 



INDEX. 



655 



Latent meanings, 2. 

Latham, R. G., diagrams, 56 

Law and logic, 7. 

Law, belief in, 625; of combinations, 
33; number of terms, 49. 

Law of contradiction, 26; and cer- 
tainty, 139; contradictory combi- 
nations, 61; for indefinite propo- 
sitions, 59; meaning of Bb, 137. 

Law of elimination, 6, 31. 

Law of, identity, 25; and certainty, 
26; applies to disjunctives, 158; 
llustration of, 50; repetition of, 
61; logical division, 27; opposites, 
25, 82; permutations, 33; relativ- 
ity, illustration of, 50; the exclud- 
ed middle, 27; for indefinite prop- 
ositions, 59; illustration of, 50; 
meaning of, 61 ; meaning of B or 
b, 137. 

Laws of thought, 24. 

Lawyers, this work for, 1. 

Leibnitz, an admirer of Lully, 611; 
sign for "some," 60; theory of 
logic, 627. 

Leechman, diagrams, 56. 

Letter, absence of, 82; "k" in 
Mnemonic lines, 362; "m" in 
Mnemonic lines, 361; "p" in 
Mnemonic lines, 361; "s" in 
Mnemonic lines. 361. 

Letters, capital, for positive terms, 
37; definitions of, 85; in two com- 
binations, reading of, 61; order 
of reading, 39; represent names 
of one idea, 49; signs for terms, 
37; small for negative terms, 37. 

Lettering diagrams, 301. 

Lewes, a logical process is a physical 
process, 642. 

Locke, John, on the syllogism, 613; 
on induction, 90. 

Logic, a graphic system of, 2; art of, 
3; Boole's method, 618; cannot 
pass on the truth of the premises, 
313; cannot solve numerical prob- 
lems, 461; concerned with words 
and thoughts only, 459; deals with 
names, 632; definition, 3; function 
of, 3; germs of, 1; problem of, 100; 
science of interpretation, 3; truth 
of the premises, 62; utility of, 601; 
when useful, 7. 

Logical, contraries, 133; equivalent 
definitions from, 133; division, law 
of, 27; existence, 458. 

Logicians, and beginners, 1; spirit of, 
639. 

Lombard, a schoolman, 609. 

Lord Karnes, on self evident prop- 
ositions, 615. 



Lotze, H., conversion of particular 
affirmatives, 416; conversion of 
particular negatives, 416; criti- 
cism of the syllogism, 314, 318, 
325; impure conversion, 415; in- 
ferring negatives, 413; on quanti- 
fication, 380; reciprocal judg- 
ments, 415. 

Lully, Raymond, author of Ars 
Magna, 611. 

Magnus A, a schoolman, 609. 

Major Premise may have four 
minors, 337. 

Mansel's Aldrich, 108. 

Mansel on "or," 205. 

Many, exclusion of, 63. 

Marquand, Dr., best diagrams, 56. 

Mathematics, definite terms, 6; dif- 
fers from logic, 3; use of signs, 
37. 

" May," meaning of, 411. 

McColl, Mr., on hypotheticals, 228. 

Meaning of, " or," 204; propositions, 
458. 

Metaphysics and logic, 7. 

Method, Cousin on, 621; for induc- 
tive examples, 538; for working 
examples, 53; of determining the 
combinations to be eliminated, 
277; finding combinations to be 
eliminated, 300; finding con- 
tradictories, 562; finding contra- 
dictories, 585; finding equivalents, 
543; finding inconsistents, 401; 
finding inconsistents, 406; this 
system, 584. 

Mill, J. S., claim for the syllogism, 
369; on extension and intension, 
21; on induction, 90; on "or," 
205; on the syllogism, 617. 

Mind, cannot be occupied with two 
things at once, 328; method of 
making a judgment, 328; nature 
of, 641; works automatically, 041. 

Ministers, this work for, 1. 

Mnemonic lines, 361. 

Modality, divisions, 21. 

Modus ponendo tollens, example of, 
184; exclusiveness of "or,' ? 185. 

Modus tollens, example in, 255, 256, 
257. 

Modus tollendo ponens, example of, 
185. 

Moods, 337; eleven allowable, 337; 
five are neglected, 338; nineteen 
have names, 338; of the fourth fig- 
ure, rules for, 359; rules for the 
legitimate, 359; the second figure, 
359; subaltern, 359. 

Most, exclusion of, 63. 

Motion, fallacy of, 600. 



656 



INDEX. 



Mutare, meaning of, 362. 

Name, called a term, 11; indicates 
existence, 10; implies an opposite, 
25; what it represents, 89. 

Names, abstract, 21 ; apply to every- 
thing, 631; collective, 12; concrete, 
21; for propositions, 246; general, 
10; general are indefinite, 10; 
general or connotative, 10; indi- 
vidual are non-connotative, 11; 
negative, conversion of, 12; rela- 
tives, 12; singular, 10; subject and 
predicate are, 60. 

Negation, implies affirmation, 623; is 
affirmation, 632; is real, 374. 

Negative, terms, elimination of, 449; 
equivalents, 376. 

Negatives, are on a par with 
affirmatives, 584; distribute pre- 
dicate, doubted, 372; meaning 
of, 12; meaning of 14; names, 
22; premises, example of, 324; 
premises, 323; propositions yield 
conclusions, 374; terms, interpre- 
tation of, 55; equivalents, 375: 
have prefix "not," 5; included in 
the contrary 636; imply differ- 
ence, 12. 

Nego, 15, 303. 

Nominalism, first taught by Antis- 
thenes, 607; doctrine of, 609. 

"No," reading of, 386, 408; transla- 
tion of figures, 164. 

Non-equivalents, example of, 470, 
472, 474, 481, 483, 485. 

"Not," a part of the predicate 
name, 370; meaning of, 375; sub- 
stitutes for, 375. 

Numerical reasoning, 3, 461. 

Object of this work, 1. 

Ob version, 412; example, 431; mate- 
rial, 426; material is illogical, 427; 
meaning of, 418, 424; names for, 
426; rule for, 412. 

Occam, W. , a Nominalist, 612. 

Old logic, limited to three terms, 
585. 

Omitting, a substitute for not, 375. 

Opposite, propositions, example of, 
455; terms, 129; opposites, 375; a 
fundamental theory, 5; incon- 
sistents, 13; law of, 25; positive 
and negative. 13; opposites, see 
law of opposites; when terms 
are, 375. 

Original matter in this work, 1. 

Order of subject and predicate, 303. 

Or, exclusion of, 159; is indefinite, 
206; meaning of, 22; sign for, 37. 

Our system, method of, 584. 

Particular, affirmatives, 303; infer- 



able from "A," 399; conversion of, 
409, 416; converted simply, 371; 
indefinite, 18; inferences from, 
397; meaning of, 15, 62; not infer- 
able from universal affirmatives, 
399; obversion, 412; negatives, 303; 
conversion of, 408, 416; inferences 
from, 397; meaning of, 15, 63; ob- 
version, 412; treated as particular 
affirmatives, 371; propositions do 
not distribute the predicate, 372; 
yield no conclusions, 374. 

Perfect contradictories, 375; mean- 
ing of, 375. 

Permutations, law of, 33. 

Petitio Princippii, fallacy of, 588. 

Philosophy and logic, 7. 

Phrenologists, best analysis of mind, 
646. 

Plato, discussed Nominalism, 610. 

Ploucquet, a symbolist, 614; dia- 
grams, 56. 

Politian, an admirer of Lully, 611. 

Political Economy and logic, 7. 

Politics and logic, 7. 

Polysyllogisms, 363. 

Positive and negative are relative, 
5. 

Positive and negative, applied to 
ideas, 5. 

Predicable, a term, 11. 

Predicables, five kinds, 11. 

Predicate, is a name, 17. 

Predicate, and predication, 17; mak- 
ing identical with subject, 305. 

Predicates, are either distributed or 
undistributed, 379; when distrib- 
uted, 372. 

Predication, 389. 

Premises, Aristotle's order, 373; 
cause elimination, 139; combina- 
tion of independent, 314; implied, 
22; imply a supposition, 138; in- 
consistency in, 82; independent, 
316; order of, 373; singular, 374; 
truth of, foreign to logic, 313; use 
of additional, 78; why so called, 
304. 

Pretended syllogism, example of, 
320. 

Prince on the Nature of Mind, 641. 

"Principles of Science," on induc- 
tion, 90, 106; solving inductive 
problems, 91. 

Probabilities, science of, 618. 

Problem, about a class of things, 
513; about a class of natural pro- 
ductions, 520; about a nation, 518; 
about buildings, 516; about mem- 
bers of a scientific society, 528; 
about three systems of proposi- 



INDEX. 



657 



tions, 523; about warm-blooded 
vertebrates, 514; from "Formal 
Logic," 526; from "Symbolic 
Logic," 522; The common council, 
506; The fishing party, 504; The 
four hunters, 505; The murder, 
500; The yacht, 501; The young 
ladies, 503; difficult, 2; method of 
solving, 480. 

Process for making subject and 
predicate identical, 420. 

Proclus, identity and diversity, 608. 

Property, a predicable, 11. 

Proposition, an act of the judgment, 
370; disjunctive, 13; meaning of, 
19; n, 379; n, 380, 392; six forms 
of, 380; two names for one object, 
370; U, 379; w, 379; w, 380; Y, 
379, 380. 

Propositions, 370; affirmative and 
negative, 14; always affirmative, 
14; A Y I, 391; consistent, 376; 
consistent and inconsistent, 6; 
consistent, what are, 376; contra- 
dictories of, 562; complex, 466; 
copulative, 18; eight forms of, 396, 
388; either premises or conclu- 
sions, 100; equivalent, appearance 
of, 166; equivalent, examples of, 
399, 453; exclusive, 19, 396; equiv- 
alence of, 399; equivalent, process 
for, 167; equivalent, Prof. Jevon's 
table, 110; form of, 19; four forms 
of, 14; have two terms, 370, 390; 
how divided, 491; how to prove 
false, 108; how to prove true, 
108; how to state, 306; iden- 
tical, 407; if false, 119; if 
true, 119; impossible, 160; incon- 
sistent, example of, 80; inconsist- 
ent, method of finding, 400; indef- 
inite, 59; indefinite, conversion 
of, 10; reform of, 59; I U n, 390; 
meaning of, 398, 458; method of 
testing, 48: prima facie meaning, 
4, 173; reading backward, 89; real, 
19; remotive 18; resemble boxes, 
166;synonymous, 36; stating of 119; 
tautologous, 36; true and false, 20; 
true, formula, 17; U and n, 395; 
universal and particular, 14; verbal 
19; what they deny, 189; when 
equivalent, 110; when inconsist- 
ent, 399; when independent, 377; 
when not true, 372, 398; when 
opposed, 370; Yand n, 394. 

Quality, divisions of, 21; doctrine 
of, 14; in a true logic, 22; of hypo- 
thetical, 224. 

Quantification, 379; example of, 386; 



42 



meaning of, 379; obtains four new 
moods, 379. 

Quantity, in a true logic, 22; of 
propositions, 14. 

Ray, on the syllogism, 360; a fallacy 
in obversion, 443. 

Reading, 286; combinations, 407; 
the subject, 62; order of letters, 39; 
"no," 408. 

Realism, doctrine of, 609. 

Reasoning, a chain of, 119; a repe- 
tition of uniform operations, 55; 
inference, 3: deductive and in- 
ductive, 6; different ways of stat- 
ing, 100; from falsity to falsity, 
125; from truth to falsity, 119; 
from truth to truth, 125; new 
system, 1; numerical, example 
of, 463; no discovery of new facts, 
100; no progress in, 100; not dis- 
cover new facts, 4; obtains other 
propositions, 100; process, 308, 31; 
to be correct, 5; process, what it is, 
308; with negative terms, 47. 

Reasoning Frame, foundation for 
logic, 51; illustration of, 142. 

Reasons why propositions are equiv- 
alents, 546. 

Reciprocating disjunctives, 223. 

Reductio, ad impossible, 444; ad ab- 

surdum, example of, 456; ad im- 
possible, 362. 

Relative and correlative, 12. 

Relation, divisions of, 21. 

Relativity (see law of); Cousin on, 
622; doctrine of, 25; fallacy of, 
596; law of, 24; Spencer on, 624; 
taught by Hegel, 616. 

Remus P. , an opponent of the schol- 
astics, 610; on laws of thought, 
612. 

Retroversion, meaning of, 419. 

Reversion, meaning of, 418. 

Roscellinus, a scholastic and Nom- 
inalist, 610. 

Rows, subdivisions of sections, 41. 

Rules, for conversion, 407; for ob- 
version, 412; obtaining the con- 
trapositive, 428; for syllogisms, 
304, 305; for the Sorites— apply 
to figure one, 366; of the Sor- 
ites, 365; of this system, results 
of, 585. 

Scholastic philosophy, 610. 

Schoolmen, list of, 609; particulars 
and universals, 609. 

Schuppe, 626. 

Scotus, D., a schoolman, 609. 

Sebonde, a follower of Lully, 611. 

Second Figure, 302, 304; example of, 
331. 



658 



INDEX. 



Sections, divisions of, 41; represent 
combinations, 49. 

Sensation of sameness is no sensa- 
tion, 24. 

Sentence, a definite, 10. 

Sextus Empericus, 608. 

Sign, for the copula, 37. 

Signs, 37; for terms, 37; for universe 
of discourse, 40; none for denial, 39; 
or, 37; usefulness of, 37. 

Sigwart, 626. 

Single terms, 127. 

Smart, example from, 272. 

Socratic logic, 608. 

"Some," a distributed term, 326; an 
inference from all, 62; inferring 
some from all, 359; means some 
only, 380; meaning of 15, 62, 380; 
means a small fraction, 635; not in- 
ferrible from all, 359; not the op- 
posite of all, 359; should be ban- 
ished, 334. 

Sorbonne, Doctors of, opponents of 
Lully, 611. 

Sorites, 365; examples of, 367; ex- 
amples of in figures Two and 
Three, 366; in Baroco, 366; in 
Bocardo, 366; rules of, 365. 

Spaulding, contraposition, 426; claim 
for the syllogism, 368. 

Species, a predicable, 11. 

Spencer, a feeling is a nervous 
change, 642; on the syllogism, 623. 

St. Anselm, an opponent of Roscel- 
linus, 610. 

St. Aquinas, on "or," 205. 

Stating propositions, 284, 306. 

St. Bonaventure, a schoolman, 609. 

Stoic, doctrine is Nominalistic, 608. 

Stoics, founded by Zeno, 607. 

Strengthened Syllogism, 359. 

Students, this work for, 1. 

St. Thomas Aquinas, a schoolman, 
609 

Studies in Logic, on Boole's system, 
383. 

Subaltern Moods, 359. 

Subalterns, 370; both may be true 
or false, 371; what are, 370. 

Sub-contraries, 370; cannot both be 
false, 371. 

Subject and Predicate, difference 
between, 385; must be identical, 
327. 

Subjects, indesignate, 16; are names, 
17; when distributed, 372. 

Subversion, 417. 

Suppressio veri, fallacy of, 590. 

Syllogisms, 302; contain, three 
propositions, 302; contain three 
terms, 302; criticised by Bain, 312; 



criticised by Brown, 617; criticised 
by Campbell, 615; criticised by 
Mill, 617; criticised by Spencer, 
623; criticised in Brewster's Ency- 
clopedia, 620; criticism of, 620; 
divided into four figures, 302; in- 
formal, 360; criticised by Lotze, 
314; criticised by Lotze, 318; criti- 
cised by Lotze, 325; criticised by 
Miss Jones, 325; criticised by 
Venn 317; defect of, 4; distinctions 
in, 39; divided into four forms, 
303; have three propositions, 337; 
imperfection of, 308: informal, 
360; informal, examples of, 368; 
imperfect, 308; Mill on, 369; Miss 
Jones' criticism of, 314; pure alter- 
native, 188; Ray on, 368; rules for, 
305; six rules of, 304; Spaulding 
on, 368; strengthened. 359; use of 
general terms, 11; Whately on, 
368; when not genuine, 374; when 
valid, 313; with four terms, 327. 

Syncategorematic, 12. 

Table, of conversion, 421; sixteen 
combinations of premises, 334; the 
four figures, 333. 

Teachers, this work for, 1. 

Tenth Amendment, 508; contradicto- 
ries to, 573; disjunctive contra- 
dictories, 583; disjunctive equiv- 
alents, 581; inconsistents, 402. 

Term, a predicable, 11; applied to 
everything, 630; combination of, 
127; common, indefinite, 59; com- 
plex, 13, 127; complex, not oppo- 
sites necessarily, 133; contraries, 
13; general are adjectives, 20; not 
contradictories, 375; representation 
of, 38; simple, 13; single, 127; sub- 
ject or predicate, 13; when op- 
posites, 375.. 

Theology and logic, 7. 

Thinking, is internal reasoning, 644; 
made visible, 1, 167. 

Third Figure, 302, 304; example of, 
330. 

Thompson, Archbishop, on the Prop- 
osition n, 393; on " or," 205; on the 
fourth figure, 362; criticism on 
the Galenian figure, 362. 

Trendelenberg, 626. 

Ueberweg, 626; equipollence, 426. 

Ulrici, 626. 

Undistributed, doctrine of, 16. 

Uneliminated combinations, 178. 

Universals, are affirmations of like- 
ness, 596; are indefinite, 635. 

Universe of Discourse, a limited one, 
101; in relation to existence, 459; 
sign for, 40. 



INDEX. 



659 



Universal affirmatives, 303; con- 
version of, 371; conversion of, 18; 
not strictly true, 17; meaning of, 
60; obversion, 412. 

Universal negatives, 18, 303; mean- 
ing of, 15, 62; converted simply, 
371; inferences from, 397; obver- 
sion, 412. 

Universal propositions distribute 
the subject, 372. 

Unless, translation of, 19. 

Utility of logic, 601. 

Unity and variety, 622. 

Venn, J., a disciple of Boole, 318; 
criticism of syllogism, 317; ex- 
ample from, 179; illustration 
from, 222; ellipses, 57; on fatal- 
ism, 596; on logical diagrams, 55; 
on "or," 204; on the contradic- 
tory of an expression, 160; pred- 
icate is a name, 372; subject and 
predicate identical, 372. 



Ventura, on analysis, 616. 

Wagner, L. H., a mathematical 
system, 615. 

Weakened conclusions, 359. 

Welton, Mr., on hypotheticals, 229. 

Whately, Archbishop, on confu- 
sion, 6; Logic, 108; claim for syl- 
logism, 368; examples in conver- 
sion, 438; example from, 280, 412; 
on "or," 205; on probabilities, 618; 

William of Champeaux, schoolman, 
609. 

Wolff C, a symbolist, 614. 

Words, are signs of thoughts, 646; 
can be reduced to nouns and 
verbs, 647; the instruments of 
reason, 645. 

Wundt, 626. 

Zeno, the father of logic, 607. 













































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